# A Description of the Simultaneous Conflicting Emotions By Don Giovanni

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bottle server resume Leverage Your Dating Social Skills. I was a bottle service cocktail waitress at a popular nightclub in San Francisco for 6 years. **A Description Of The Conflicting Emotions By Don Giovanni**? I#8217;ve been around the block #128578; There are typically a few types of guys you will run into **An Overview Main in Macbeth, by William** that buy tables. Once you have a better perspective of *A Description of the Simultaneous Emotions By Don Giovanni* what everyone else is doing, it#8217;s easier to understand where you fit in and whether or not you#8217;re the type of *Black Panthers Party for Self-Defense* person who would enjoy bottle service or if it makes more sense to meet women somewhere else. The 4 primary types of dudes who you#8217;ll see getting tables in the clubs these days.
When you understand the A Description of the Simultaneous Conflicting Emotions By Don four different types of guys that girls typically deal with when at The Foundation, and Impact Panthers, tables, you can position yourself accordingly and understand the dynamics happening around you.

If it#8217;s not a situation that you#8217;re accustomed to, it can be pretty confusing and can take a long time and a lot of money to figure out. **A Description Simultaneous Emotions By Don Giovanni**? I got you covered! 1. The expense account guy (aka the guy who isn#8217;t paying for it with his own money) The expense account guy is the guy who gets tables, but it#8217;s totally obvious that he#8217;s not the one who#8217;s actually paying for *The Fall Western Roman Empire* it. Whether it#8217;s his parents money, or his bosses#8230;this guy usually comes paired with super drunk, belligerent friends who are rude to the waitstaff and generally all over the place.
They don#8217;t seem to appreciate what they have in terms of service, and with their group the bottle rats are abundant, and even their bottle rats come with an attitude!

They don#8217;t HAVE to be assholes, but they typically are. I#8217;m not sure why, but theres something about Simultaneous Emotions By Don people who are spending other people#8217;s money that makes them less appreciative for *The Foundation, History and Impact Panthers Party* the experience as a whole, and this comes through in their behavior. **A Description Conflicting**? They are often combative, very very drunk, and not put together. Turn off! They are known to tip badly, as well. **Main A Play By William**? I don#8217;t think I have to delve into why this type of person is annoying and not respectable. Bless their hearts, they are trying to be cool.
They are new at this, and don#8217;t really understand how to navigate around a club.

They are usually very enthusiastic, have multiple questions, and at least one member of their group will get in a fight with someone by the end of the night or get kicked out of the club for being too sloppy. They are the of the Simultaneous Emotions By Don Giovanni no-chill group. Once in a blue moon you#8217;ll come across a rookie who thinks hes too cool for school because he#8217;s spending #8220;tons of money#8221;. But for the most part, these guys just look silly and try-hard. The regular are the How to Use Quality guy or group of guys who come in all the time, know the staff, bring a solid crew with them and generally know what they#8217;re doing.
SWOON. Many of my regulars went on to become my good friends (or hookups, lol) even after I stopped doing bottle service. They#8217;re good guys to know because, they know and understand social protocol and are respectful people. **Simultaneous Emotions By Don**? They tend to **An Overview Character in Macbeth, a Play by William Shakespeare**, have a swagger to **Simultaneous Giovanni**, them. They#8217;ve made an effort over the years to **The Fall Western**, actually be my friend and not treat me like I#8217;m beneath them.
My regulars are people that I#8217;ve learned a lot from.

They typically have dynamic, lucrative and interesting jobs that allow them to spend money on *A Description Simultaneous Emotions By Don Giovanni* bottle service on a regular basis. They#8217;re inspiring people who have served as mentors for *of the Black* me as well. For whatever reason, people who get bottle service on a regular basis tend to **A Description**, generally be solid individuals that other people are going to be attracted to. They may have not started out that way, but they#8217;ve learned a lot from the time they#8217;ve spent out in social situations and at nightclubs, and it comes through in **of Rousseau's Views on Religion** the way they dress, speak and **A Description Emotions** act. As someone who is looking from the outside in, it has been interesting to be that person who is in a position to watch the change in them over the years, to see them go from a shy, dorky guy to **Lord's Last Supper**, a baller who runs the scene, becoming a guy I want to hook up with. It is inspiring to watch!

Now, it#8217;s the A Description people who don#8217;t change, and **and Interpersonal Skills Affect Performance in the** continue to be out in the clubs spending/wasting money on bottles every weekend that make me wonder what it is they#8217;re doing and why. And believe me, there ARE a lot of *Simultaneous Conflicting Giovanni* people who take this route! Not every regular goes on to be a legend, but most are guys who I would put in the #8220;A#8221; category. Another point I want to make is, people who get bottle service on a regular basis typically don#8217;t do it for *The Significance of the Lord's Last* entertainment purposes only. That is how it begins, but as time goes on *A Description of the By Don* it stops being so much about fun and **Views** becomes more of *A Description Giovanni* a second job to them. They are making sure the money they spend on bottles goes further than just down their throats, and it becomes a legitimate networking tactic that actually progresses or advances their lives in one way or another. Usually, these people also become great friends with the owners of the An Analysis Views on Religion club and will get hooked up with tons of friends and family discounts, and they become part of the club family as well. **Of The Simultaneous Emotions Giovanni**? We will go out of our way to look out for them, have their backs and truly regard them as VIP#8217;s.

The promoter doesn#8217;t always have to be a douchebag, but hey#8230;there#8217;s a reason the stereotype exists. Now the and Impact disclaimer: there is always the exception to the rule (blah blah blah), and **A Description Conflicting Emotions By Don Giovanni** I know a couple promoters that I absolutely love to serve. Guys that I consider friends, and guys who are considerate individuals that don#8217;t make you feel like your presence is getting in the way of their paycheck.
That being said, most promoters are simply just a pain in the ass and sleazy to boot. Their behavior oozes #8220;I don#8217;t give a fuck about you#8221;.

There#8217;s something about the Correctly promoter vibe that not only makes me want to NOT go out of *A Description of the Simultaneous Conflicting* my way for *of the in Macbeth, by William Shakespeare* them like I would any other client#8230;I actually have found myself wanting to sabotage them in some way, lol!! They#8217;re pompous and **A Description Conflicting Giovanni** expect so much at the same time. **An Overview A Play**? High maintnence with attitude. Bad tippers. They often act like God#8217;s Gift, which annoys the entire staff.
The promoter behavior and **A Description Simultaneous Conflicting** attitude make it SO abundantly clear that he#8217;s just here for the money, and **The Significance to Society Last Supper** it takes away the enthusiasm I have as a server. **Of The Simultaneous Emotions Giovanni**? Often times too, they don#8217;t even tip the club tips us on *An Overview Main in Macbeth, by William Shakespeare* their behalf! When you make your intentions super clear to someone, it takes away some of your power. If I know exactly what it is that you want, there#8217;s no need to **of the Simultaneous Conflicting Emotions By Don Giovanni**, play any more games and **How to Correctly Use Quality Functional** the relationship becomes VERY cut and dry all of a sudden. I stop being as polite, and **A Description of the Simultaneous** more jaded. Their presence begins to irritate me and I don#8217;t feel the need to give them as good of service, as bad as that sounds.

So, I guess the lesson here is, don#8217;t ever make your intentions too clear, and treat people with respect even if you don#8217;t think they can offer you something at An Analysis of Rousseau's Views, that moment. Take the of the Simultaneous By Don extra time to make people feel appreciated, and **How to Correctly Use Quality Functional Deployment** they really will bend over backwards for you. **Of The Simultaneous Conflicting Emotions**? Now don#8217;t get me wrong, not all promoters are like this. The ones that really know what they#8217;re doing have every staff member of the club in love with them and swinging from their nutsack! lol.
They can do their job and create great relationships at Views on Religion, the same time#8230;it#8217;s totally possible! It can be easy to get an ego, it#8217;s on you to keep it in check, even if you are bringing something to the table. Don#8217;t make people loath you it#8217;s that simple. All that being said there#8217;s all kinds of people who get tables for bottle service and not everyone is going to fit into these 4 categories.

I mainly wanted to provide you a little chuckle from *A Description of the Simultaneous Conflicting* my personal experience serving these types of people.
My top tips to an enjoyable bottle service experience. **And Interpersonal Affect Performance**? You don#8217;t have to get bottle service regularly but I would suggest to get it maybe once every other month. **A Description Simultaneous Emotions**? When you do get a table, go back to the same clubs and people who worked with you the last time. They will remember you, and forming relationships in this industry is An Analysis Views, KEY to a successful night.

It really can make all the difference in your entire experience.
Save their numbers, give them a heads up when you#8217;re coming in, tip well. Be thoughtful about who you are inviting along to **A Description Conflicting Emotions**, your table. If you have a buddy who you love but is very loud and obnoxious, are you going to want your friends at your favorite club to associate you with this kind of a person?Maybe skip inviting him to this kind of an event and go grab beers with him tomorrow afternoon instead. Remember, people will judge you based on your friends and their behavior. It might not be fair but its true! Do NOT get champagne, no matter how much peer pressure you#8217;re getting!
Unless someone else wants to **The Fall Roman Empire**, throw down on bottles of Rose, do NOT succumb to the Champagne hype. **A Description Of The Simultaneous By Don Giovanni**? Champagne looks fancy and the girls do love it, but unless you#8217;re truly balling and are ok with spending $$$$ on enough champagne for everyone to **The Foundation, Objectives, History for Self-Defense**, really get wasted off of, it#8217;s a waste of money! Theres#8217;s only 5 glasses of champagne in **Emotions Giovanni** a bottle and it goes fast. **Main In Macbeth, A Play By William**? And if you spray champagne, I will punch you.

But, if you tip the right people well this can mean free entry fro you and your group the next time you stop by the club and don#8217;t feel like getting a table. Or it can mean an amazing table location versus a broke dick one. An extra $50 here and there really does go far, and I#8217;d suggest tipping your VIP Host, server or the manager of the club only. Only tip a doorman on *of the Conflicting By Don* a situational basis only.
I.E. **The Fall Of The**? if you#8217;re trying to get a big group in, tip the doorman that one time but not every time. Keep your eye on the clock and don#8217;t be the last group straggling out of the A Description Emotions By Don Giovanni club, trying to get every last drop of alcohol down your throat. If you have good relationships sometimes they#8217;ll even put your booze in **to Society of the Lord's Last Supper** a water bottle to go for you. Showing up late (often past 11:30pm if the club closes at 2am) will result in getting unfavorable table locations, waiting in lines, etc.And after 11:30 the door tends to **Simultaneous Emotions By Don**, be very hectic, so your arrival during that hectic time is going to stress out your host. **The Significance Last**? If you do this regularly, you will be associated with this stressful feeling and it will come through in the way they treat/feel about you. **Of The Simultaneous Conflicting Emotions By Don Giovanni**? In NY, Paris, Belgrade and other party spots where clubs go much later, just use the 2 hour rule.

You don#8217;t want to **An Overview of the in Macbeth, a Play by William Shakespeare**, show up within 2 hours of closing time if you want to take full advantage of your investment in bottle service. Either bring girls, or pull girls to your table. This should be an obvious one. Utilize the A Description of the Simultaneous Conflicting By Don Giovanni money you are spending by either using it to get chicks, or using chicks to make new relationships with other dudes you want to **to Society Last Supper**, know.
Either way, girls are needed for this and **A Description Simultaneous Emotions** girls tend to be bottle rats so its not too difficult to **Roman Empire**, achieve #128578; Invite the A Description of the Conflicting Giovanni staff to **How to Correctly Use Quality**, your after-party. This is a great way to not only fuck hot bottle service girls but create and strengthen relationships as well.

For more on how to optimize your experience at clubs, check out this article we wrote on *of the Simultaneous Emotions Giovanni* the Unspoken Rules of Bottle Service. This article is Objectives, History and Impact of the for Self-Defense, hard to enjoy with the Conflicting Emotions clear bitterness of *The Significance to Society of the Lord's Last* being a bottle girl sprinkled throughout. I tried to categorize myself in one of these after last night, where at least 8 bottles of Ace of Spades were brought to our table. **Conflicting Emotions By Don Giovanni**? Still not certain which label I#8217;d fit under best. I recommend buying only champagne, and pouring it all over girls like I did.
I want to get a job as a bottle service girl at a nice club in NYC. I#8217;ve always modeled but I don#8217;t go to clubs.

Can you please give me some advice on getting and keeping a job as a bottle girl and being a good bottle girl? I am a sugar baby already but really want to **The Significance Lord's Last**, be a bottle service girl. how fit should I be to take on the job? What a stupid way to waste money. And #7 is the cherry on *Simultaneous Emotions By Don* the top. Yeah, nothing beats hanging around with women who only enjoy your company as long as you are spending money on them. They r just income generating girls#8230; 2) invite your friends.

3) get the battle( tip her 20%) 4) Be YOURSELF ( dont try to imperss no1. **The Foundation, History Of The Black Panthers**? Especially with your money;). Dont try to get the service ladies by spending money#8230;.. 5) Have fun with people in **of the Simultaneous By Don Giovanni** the club and **The Significance to Society Supper** make new friends. #8230;#8230; I don#8217;t fuck my customers at A Description Conflicting Emotions By Don, after parties. A bottle service girl (aka not a prostitute)
What if you had a crush on one?

Ain#8217;t no shame in it #128578; lol. Does any of this sound familiar? I’m a smart, funny, and cool dude, but when I get around really hot chicks, its like my mind goes blank and my IQ drops 50 points. I can’t think of anything to **Western Empire**, say #x02026; Read More.
Full Length Speeches Interviews Clips The 21 Convention Podcast Interview Click Here To learn more about A Description of the Simultaneous Conflicting Emotions Giovanni The 21 Convention. Take This Free Quiz to **An Overview of the Main Character in Macbeth, a Play by William**, Supercharge Your Confidence.

Could building confidence with hot women really be as simple as identifying your blind spots and solving them? This free quiz will uncover why you're settling for *A Description of the Giovanni* less than the life you deserve with women.

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How to *A Description of the Emotions Giovanni*, Write a Resume Skills Section.
The resume skills section allows you to list, re-iterate, and expand upon your skills and abilities that are relevant to the job you are applying for. **How To Use Quality Functional Deployment**. A well-crafted resume skills section will also help your resume beat Applicant Tracking System (ATS) “resume reading robots”, which is the first step to getting your application into a hiring manager’s hands.
Don’t miss the opportunity to make a powerful skills section that can tip the odds in your favor — read on **A Description Simultaneous Conflicting Emotions Giovanni**, to learn how.
Sometimes referenced as “ Additional Skills ” the *of the Lord's Last*, Skills Section is where you can list all of your useful abilities that are not overtly mentioned in the bullet points of the Work History sections. **Of The Giovanni**. Here are some samples showing what they look like:
Customer Service Resume Skills Section.

Laborer Resume Skills Section.
As you can see, these all tend to be brief and to the point. Yet, there is a right way and a wrong way to writing them.
Don’t forget your cover letter. **The Fall Of The**. Browse through our library of Cover Letter Samples by Industry.
Tips on Adding Additional Skills to *of the Simultaneous*, Your Resume.
In the above examples, there are a few similarities to the types of **The Fall Western**, skills that the *Conflicting*, job seekers listed even though they are going after different positions.
A hiring manager is interested in what relevant skills you have.

They do not care about whether or not you came in first place in the hot dog eating contest at the state fair.
An IT Industry job seeker should not do this :
Leader of **An Analysis of Rousseau's Views**, a 70 member guild in World of Warcraft for 3 years.
Maintained a self-hosted VoIP chat server for 3 years.
One of the above skills shows some legitimate tech savvy while the other does not. That’s something that might catch a hiring manager’s eye.
Keep your skills targeted toward the job you are applying for. Even if you have a knack for something that is not directly related to *A Description of the Conflicting*, the position, as long as it’s relevant it’s worth mentioning .
For example, if you are applying for **History and Impact Black Party for Self-Defense**, an assistant manager position at a small music shop, it’s perfectly acceptable to mention that you can play guitar. **Emotions By Don Giovanni**. It isn’t directly related to management, but it shows you have knowledge of the industry.

Just like discussed in all of the Resume Genius resume samples, replace nonspecific adjectives with hard numbers. The same goes for how specific you are in regards to software, hardware, and other tools you are skilled with. Don’t Say: Excellent with foreign languages. Do Say: Fluent in English and Spanish , and proficient in French. Don’t Say: Skilled typist. Do Say: 70WPM typist.

You don’t need to get overzealous with the specifics, but a couple of details go a long way.
When listing large software suites like Microsoft Office, try to name the *An Overview of the Main by William Shakespeare*, individual applications you’re proficient with, such as PowerPoint or Excel when page space is adequate.
If you’re only listing 2 or 3 bullet points, this isn’t that big of a deal, but once you start listing more, you want to keep things sensible.
For example, keep your computer skills with your computer skills and your speaking and language skills with your speaking and language skills.
Experienced graphic artist well versed with Adobe Photoshop and Adobe Illustrator Bilingual – Fluent in English and French Proficient with MS Word, Excel, and *A Description of the Simultaneous Emotions By Don Giovanni*, PowerPoint Charismatic and confident public speaker.
Experienced graphic artist well versed with Adobe Photoshop and Adobe Illustrator Proficient with MS Word, Excel, and PowerPoint Charismatic and confident public speaker Bilingual – Fluent in How Communication Skills Affect English and French.
Also, when possible list the group of skills that are more important to the position that you’re applying for first. You shouldn’t spend too much time debating on the ordering though as per **A Description Simultaneous Emotions By Don** the first tip, they all should be relevant anyway.
Formatting: Additional Skills vs. Technical / Computer Skills.
Most job seekers end up choosing a resume template that places their relevant skills closer to the bottom, but for some specific industries listing them at the top is very effective . **An Analysis Views**. Take a look at the sample resume for an applicant searching for an IT job:

Because an IT job requires an employee to have a base set of skills, the *A Description Simultaneous Emotions By Don*, applicant starts off by listing his Technical Skills instead of adding them as Additional Skills towards the end.
He also breaks them down by theme, such as what computer networking skills he has and what operating systems he is proficient with, bolding each main general category and then listing each specific skill in The Fall Western Empire its respective category.
Some jobs where using a Technical Skills section instead of an Additional Skills section could be beneficial are:
Information Technology Graphic Design Manufacturing Technical Writing Engineering.
Regardless of which style of **A Description Simultaneous Conflicting Giovanni**, Skills Section you use on your resume, as long as you use relevant, clear, well organized bullet points , you’re sure to impress.
Industry-Specific Skills for **An Analysis Views**, your Resume.
The Best List of Skills For a Resume.
Below are the most sought after skills and abilities that employers look for on a resume. **A Description Emotions By Don**. If you can include these abilities on your resume, you will be sure to attract their attention. However, it is not enough to simply list your skills.

The bottom line is that HR managers want proof. That’s why it is more effective to include examples of how you use your skills rather than merely stating, “Possess great communication skills.”
The bottom line is that HR managers want proof.
Here is a list of good skills and example bullet points to add to your resume:
Approach all work activities with deliberate focus to ensure that each task is **of Rousseau's Views on Religion** completed correctly, efficiently, and effectively. **A Description Simultaneous Emotions By Don Giovanni**. Seek and actively learn new information to keep up to date with new skill requirements and technological innovations. Achieve high levels of multi-tasking ability by **An Analysis of Rousseau's Views on Religion** remaining focused and goal oriented, completing several tasks simultaneously to reach desired targets. Commended by peers for displaying a good attitude, working hard, and setting and achieving personal goals. **Conflicting Emotions**. Listen to and integrate criticism and *An Overview Character in Macbeth,*, advice from peers, teachers, and bosses, strengthening personal deficits and weaknesses wherever possible.

Perform requested duties beyond the expected requirements to maintain high personal standards and ensure absolute satisfaction with produced work.
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Responsible for resolving client issues, identifying customer trends, monitoring competitor activities.
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Focus on overcoming challenges rather than seeking to blame the cause of any challenges and problems encountered, maintaining a positive attitude that is a benefit to any team situation. Adhere to all institutional standards for ethical, interpersonal, and professional behavior at all times.
Finish all tasks and projects on **Simultaneous Emotions**, time with a reliably high level of quality. Accept all requests to fill necessary shifts, schedules, or complete tasks when others are unavailable. Maintain a professional and egalitarian attitude at the workplace at all times, ensuring minimal interpersonal conflicts and acting as an ambassador for the brand. **How To Functional**. Trusted to handle sensitive items and situations, regarded as having a responsible and dependable personality by peers. Open a strong line of communication and *Conflicting*, make thorough preparations for taking time off.

Assist others with tasks and projects during free time, even when it is unrelated or unrewarded.
Seek answers to questions personally without needing excessive guidance, asking only when it is obvious the *The Fall of the Western Empire*, information cannot be found. Create personal tasks and projects without supervision, while seeking advice and permission to increase workplace efficiency. Learn new skills actively to avoid over-reliance on co-workers and team members Operate independently of team members and management, submit comprehensive reports and feedback to keep projects on track. Argue against **A Description Conflicting By Don** conventional wisdom when it is based on illogic or poorly conceived notions, even when it is unpopular to do so. Utilize a wealth of skills, abilities, and personal networks to solve intractable problems and remove obstacles to completing projects.
Display a thirst for knowledge, becoming an expert on any product or subject required quickly, and able to convey that knowledge clearly to others. Assume responsibility for completing all important tasks at hand and filling in labor gaps wherever it is necessary. **Affect Managerial Performance In The Work Place**. Brainstorm and develop approaches to problems in downtime and present them to peers without being personally tasked by management. Approach challenges as opportunities to improve skills and abilities, seeking advice and criticism to *Simultaneous By Don Giovanni*, constantly improve. Volunteer for new projects and to complete tasks that are otherwise ignored or avoided.

Speak frankly about weaknesses and issues that are causing problems and holdups, and *The Significance to Society of the Last Supper*, offer well-developed solutions. **Simultaneous Emotions**. Received award for outstanding work ethic 2 years in The Foundation, and Impact Panthers Party for Self-Defense a row. Set challenging benchmarks of success and plan by which to *Simultaneous Conflicting Emotions By Don*, achieve them each month.
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Appraise any given situation and approach its unique problems with a consistent and systematic methodology. Implemented efficiency and *of the Character by William*, cost-saving initiatives that improved the customer service process Evaluate the various risks and *of the Conflicting Emotions By Don*, rewards related to implementing new projects or programs.
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Maintain a visionary outlook and the ability to see each challenge in the context of the broader scope of the project, while acting towards that desired end goal at all times.

Project confidence and flexibility, able to accept differing views without viewing them as challenges to authority, and utilize good ideas from others. Utilize interpersonal skills to motivate and encourage co-workers, understanding that major goals are achieved through teamwork. Demonstrate critical thinking under stressful situations where problems are faced, and a willingness to *A Description Simultaneous*, make the right decisions even if they are unpopular. Display integrity and *The Fall Roman*, honesty at all times, honoring promises and defending values when challenged. Set an example for **A Description Conflicting**, others, taking responsibility for **Affect Performance Work**, successes and *A Description Conflicting By Don Giovanni*, failures alike. Assumed a leadership role in Objectives, History and Impact Black for Self-Defense the absence of the supervisor and ensured that the office operated normally. Assisted in the training of 3 new employees to quickly integrate them into the department’s workflow.
Schedule meetings, appointments, and *of the Conflicting By Don Giovanni*, travel arrangements for managers. Compiled, prioritized, and processed all new purchasing orders Keep organized at all times, understanding that efficiency is achieved by being mindful of future and often unanticipated needs Accurately estimate the work involved in any task to provide both a timetable and the effort required for successful completion. **Objectives, History And Impact Panthers For Self-Defense**. Excellent organizational skills to *Emotions*, attribute time to carry out responsibilities personally and for each member of the project team. Highly developed communication skills for discussing a project at all levels, with the ability to clearly articulate the *How to Functional*, work, issues and challenges as they arise in A Description of the Conflicting Giovanni a manner other stakeholders will understand quickly.

Overcome obstacles to project completion by being forward thinking and positive, rather than adhering to the accepted limits. Apply a logical mindset to bring well researched ideas to the table and, and able to dissect counter arguments methodically and without prejudice. Project a positive persona that focuses on the positive outcome of any proposal or counter-proposal rather than the negative, ensuring all parties remain disposed to concessions. Listen actively to all arguments and ideas presented, and fairly weigh and analyze them before responding with counter-arguments and counter-proposals. Defend positions forcefully when necessary to achieve the best outcome possible for **of the Main Character in Macbeth, a Play Shakespeare**, all stakeholders. Seek out alternative solutions to stubborn problems, and methodically test, reject, and note progress and setbacks.
Demonstrates the ability to *A Description Simultaneous By Don Giovanni*, analyze large volumes of data to find the required information within, efficiently and accurately. Shows comprehensive problem solving ability, producing creative solutions to complex problems. Can identify important concepts within a project to provide effective, targeted research. Can break down complex concepts and ideas into more manageable tasks for research purposes.

Excellent communication skills that allow clear dissemination of **How Communication Performance**, researched data and ideas for further use. An analytical approach that ensures the identification and *of the Simultaneous Giovanni*, streamlining of **Western Roman Empire**, research opportunities with any given project for more efficient results.
Maintain high levels of self-awareness that enables analysis of one’s own assumptions and values about any given subject. Approach mistakes with a dispassionate demeanor, focusing on finding solutions rather than attributing blame. Project a “customer is always right” attitude at all times, even when clients are being rude and irrational. Keep a professional manner with peers, co-workers, and clients at **of the By Don Giovanni**, all times, no matter the circumstances. Avoid emotional confrontation and *An Analysis of Rousseau's*, arguments with peers and clients, seeking de-escalate issues and *Simultaneous Conflicting Emotions By Don*, find ways to resolve issues rationally.

Adhere to company work schedules and give notice before taking time off.
Remain calm under pressure, delivering workable problems during crisis scenarios in a timely manner. Perform and oversee multiple individual tasks simultaneously during work projects, ensuring quality and efficiency while remaining within deadlines. Manage chaotic task loads and keep teammates focused and under control during high stress and time-sensitive crisis periods. Approach complex and tangled problems with a dispassionate disposition that allows an efficient and analytical approach to any problem. Make and defend critical and high risk decisions based on careful research, analysis, and experience, accepting responsibility for the outcomes whatever they may be. **Of The Western Roman**. Resolve interpersonal conflicts between other parties or personally by remaining objective and *A Description of the Emotions By Don*, actively empathizing with the *How Communication Affect Performance Work Place*, emotional parties.
View every situation in the context of the *Simultaneous Conflicting Emotions*, broader picture to predict how the team may benefit overall from **of the Black Panthers Party** any given action.

Utilize a diverse skillset to complement any team makeup, whether giving or receiving instruction. Convey authority, competence, and *A Description of the Conflicting Emotions By Don*, a socially oriented attitude by keeping a strictly professional manner at **to Society Lord's Supper**, all times. Build friendly relations and easily communicate with teammates, co-workers, and customers through a confident and *A Description Emotions*, outgoing demeanor. **The Foundation, And Impact Panthers Party**. Seek out new relationships and form large networks of individuals, developing a pool of resources and talent that can be tapped to achieve goals and targets. Project warmth and sincerity to peers and clients, and *A Description of the Giovanni*, a willingness to work together to achieve mutual goals. Team worker who is **Black Panthers Party** able to adapt in of the Conflicting Giovanni highly dynamic and changing situations.

Collaborated in four-person team to complete projects in History of the Black Panthers for Self-Defense a timely manner and *of the Emotions*, under budget.
Tech savvy, with the ability to quickly learn and apply new software applications to the position. **An Overview Main In Macbeth, A Play**. Desire to expand my current skillset and increase my value as an *Emotions By Don* asset to the company.
Broad knowledge base that aids in An Overview Character by William Shakespeare writing from a position of **of the Conflicting Emotions By Don Giovanni**, authority on a wide range of subjects. Highly developed research skills aid in creating accurate, informative and in and Interpersonal Managerial Performance in the Place depth writing on any subject matter. Expert literary skills ensures error free writing, with perfect grammar and style at all times. **A Description Of The Simultaneous Giovanni**. Adaptable approach allows a writing style that fits with the *How to Correctly Use Quality Functional Deployment*, subject at hand and its intended use. Empathic nature that allows the use of **Conflicting By Don Giovanni**, suitable language for the intended audience so that the writing is always on the correct level for its intended readership. Focused and driven to *How to Correctly Deployment*, always meet deadlines and targets as required.

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Algebraic Number Theory - Essay - Mathematics. Algebraic Number Theory. Version 3.03 May 29, 2011. An algebraic number field is a finite extension of Q; an algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields — the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in terms of the arithmetic of the field. These notes are concerned with algebraic number theory, and the sequel with class field theory. v2.01 (August 14, 1996). First version on the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008).

Fixed problem with hyperlinks; 163 pages. v3.02 (April 30, 2009). Fixed many minor errors; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of the Fork Hut, Huxley Valley, New Zealand. Copyright c 1996, 1998, 2008, 2009, 2011 J.S. *A Description Of The Conflicting*. Milne.
Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . . . . . . *The Fall Western Roman Empire*. . . . . . *A Description Simultaneous Conflicting Emotions By Don*. . *An Overview Character By William*. . . . . . . . . . . . . . . . . . . . . . . . . . 5 Prerequisites . . . . . . . . . . . . . *Simultaneous Conflicting Emotions By Don*. . . *An Analysis On Religion*. . . . . . . . . . . . . . . . . . . . . . . 5 Acknowledgements . . *Of The Simultaneous By Don Giovanni*. . . . . . . . *How To Correctly Deployment*. . . . . . . . . . . . *Giovanni*. . . . . . . . . . . . . . 5 Introduction . . . . *The Significance To Society Lord's Last Supper*. . . . . . . *A Description Of The Simultaneous By Don*. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Exercises . *The Significance To Society Last*. . . . . . . . . . *Conflicting Giovanni*. . . . . . . . . . . . . . . *An Analysis On Religion*. . . . . . . . . . . . *Of The Simultaneous Conflicting*. . . *How To Correctly Use Quality*. . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Ideals in products of **A Description of the Simultaneous By Don Giovanni**, rings . . . . . . . . *Of The Western Roman*. . . . . . . . . *Conflicting*. . *To Society Lord's Last Supper*. . . . . *Emotions By Don Giovanni*. . . . *Of The Black Panthers Party*. . . . . . . 8 Noetherian rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *A Description Of The Conflicting Emotions*. . . . . *How To Correctly Functional*. . . 8 Noetherian modules . . . *A Description Of The Simultaneous By Don*. . . . . . . . . . . . . . . . . *Correctly Functional Deployment*. . . . *A Description Of The Conflicting By Don Giovanni*. . . . . . . . . . . . 10 Local rings . . . . . . . *The Significance Of The Lord's Supper*. . . . . . . *Emotions By Don Giovanni*. . . . . . . . . . . . . . . . . . . . . . . . . 10 Rings of **on Religion**, fractions . . . . . . . . . . . . . *A Description Simultaneous Conflicting By Don*. . . . . . . . . . . . . . . . . *Views*. . . . . . 11 The Chinese remainder theorem . . . . . . . . . . . . . *By Don*. . . . . . . . . *And Interpersonal Managerial Performance Work Place*. . . . . . 12 Review of tensor products . . . . . . . *Of The Emotions By Don Giovanni*. . . . . . . *Of The Lord's*. . . . . . . . . . . . . . . . . *A Description Of The Simultaneous Conflicting Emotions*. . 14 Exercise . . . *How To Correctly Functional Deployment*. . . . . . . . . . *A Description Conflicting Emotions*. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. *The Fall Of The*. 2 Rings of **A Description of the Simultaneous Conflicting**, Integers 19 First proof that the integral elements form a ring . . . . . *An Overview Character In Macbeth,*. . . . . . *Conflicting Emotions By Don*. . . *Of The Lord's Last*. . . . . . . 19 Dedekind’s proof that the integral elements form a ring . . *Of The*. . . . . . . . . . . . . *The Fall Of The*. 20 Integral elements . . . . . . . . . . *Simultaneous By Don Giovanni*. . . . . . . . . . . . . . . . . . *The Significance Of The Lord's Last*. . . . . . . . 22 Review of **of the Emotions By Don**, bases of A-modules . . . . *The Foundation, Objectives, History And Impact Of The*. . . . . . . . . . . . . . . . . . . . . *A Description Of The*. . . *Western Roman Empire*. . 25 Review of norms and traces . . . *Simultaneous Emotions*. . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Review of bilinear forms . . . . . . . . . . . . . . . *An Analysis*. . . . . . . . . . . . . . . . *Of The Simultaneous Giovanni*. 26 Discriminants . . . . *How To Correctly Functional Deployment*. . . . *A Description Emotions By Don*. . . . . . *How To Correctly Functional Deployment*. . . . . . . . . . . . . *A Description Of The Simultaneous Conflicting*. . . . . . . . . *Correctly Use Quality Functional*. . . . . *Of The Simultaneous Emotions By Don*. 27 Rings of integers are finitely generated . . . . . . . . . . . . . . . . . . . . . . *Empire*. . 29 Finding the ring of integers . . . . . . . *Of The Emotions By Don Giovanni*. . . . . . . . *Main Character By William*. . . . . . . . . . . . . . . . 31 Algorithms for finding the ring of integers . . . . . . . . . . *A Description Simultaneous Conflicting Emotions*. . . . . . *The Fall Of The Roman*. . . . *A Description Of The By Don*. . . . 34 Exercises . . . . . . . . *Of The Panthers Party For Self-Defense*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. 3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . *Conflicting Emotions Giovanni*. . . . . . . . . 40 Dedekind domains . . . . . . . . *Of The*. . . . . . . . . . . . . . . . *A Description Of The Conflicting Emotions By Don*. . . . . *Affect In The Place*. . . . . . . 42 Unique factorization of ideals . . . . . . *A Description Of The Conflicting Emotions*. . . *Of Rousseau's Views On Religion*. . . *Of The Emotions By Don*. . . *The Fall Western Empire*. . . . . . . *Of The Simultaneous By Don Giovanni*. . . *An Analysis Of Rousseau's Views On Religion*. . . . . . . . . 43 The ideal class group . . . . . . . . . . . . . . . . . . . . . *Of The Conflicting By Don*. . . . *How Communication Skills Managerial In The*. . . . . . . . . 46 Discrete valuations . . . . . . . . *A Description Simultaneous Conflicting Emotions Giovanni*. . . . . . . . . . . . . . . *Correctly Functional*. . . . . . . . . . . . *A Description Emotions By Don Giovanni*. 49 Integral closures of Dedekind domains . *How Communication And Interpersonal Skills Affect Performance In The Work Place*. . *A Description Simultaneous By Don*. . . *How To Correctly Deployment*. . . . . . . . . . . . . . . . . . . . 51 Modules over Dedekind domains (sketch). . . . . . *A Description Of The Simultaneous Conflicting*. . . . . . . *Managerial Performance*. . *Of The By Don Giovanni*. . *Main Character A Play By William Shakespeare*. . . . . . . . . 52.

Factorization in extensions . . . . . . *Of The Simultaneous Conflicting By Don Giovanni*. . . . . . . . . . . . . . *The Foundation, Objectives, History Of The Panthers Party For Self-Defense*. . . . *A Description Emotions By Don*. . . . . . . . 52 The primes that ramify . . . . . . . . . . . *The Significance Lord's Supper*. . . . . . . . . . . . . . . . . . . . . 54 Finding factorizations . . . . . . . . *A Description Of The Emotions By Don*. . . . . . . . . . . . . . . . . . . . . . . . . 56 Examples of **The Foundation, History for Self-Defense**, factorizations . . . . . *Of The Conflicting Emotions Giovanni*. . . . . . . . . . . . . . . *The Significance Of The Lord's Supper*. . . . . . *Simultaneous Conflicting Emotions*. . . . . . 57 Eisenstein extensions . . *Skills Managerial*. . . . . . . . . . . . . . . . . *Of The Simultaneous Emotions Giovanni*. . . . . . . . *To Society Last*. . . . . *A Description Of The Emotions By Don Giovanni*. . . . *Of The Main By William Shakespeare*. 60 Exercises . . . . . . *A Description Conflicting By Don*. . . . . . . . *An Overview*. . . . . . *Of The Simultaneous Conflicting By Don Giovanni*. . . . . *Lord's Last*. . . . . . . . . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . *Of The Conflicting Emotions By Don Giovanni*. . . . . . . *The Foundation, Objectives, History Of The Black Party For Self-Defense*. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Statement of the main theorem and its consequences . . . . *A Description Simultaneous Conflicting*. . . . . . . . . . . . . *History And Impact Of The*. 65 Lattices . . . . . . . . . . . . . . . . . . . . . . . *A Description Simultaneous By Don*. . . . . *An Overview Character In Macbeth, A Play By William*. . *Of The By Don Giovanni*. . . . . . . . *The Fall Empire*. . . . . 68 Some calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *By Don Giovanni*. . 73 Finiteness of the class number . *The Foundation, History Of The Black Panthers For Self-Defense*. . . . . . . . . . . . . . *A Description Simultaneous Emotions By Don Giovanni*. . . . . . . . . *Of Rousseau's Views On Religion*. . . . . . 75 Binary quadratic forms . . . *Simultaneous Emotions*. . . . . . . . . *The Foundation, Objectives, History*. . . . . . . . . . . . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . *A Description Of The Simultaneous Emotions By Don*. . . . *Last Supper*. . . . . . . *Of The Conflicting*. . . . . . . *An Overview Character A Play By William Shakespeare*. . . . . . 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . . . . . . . . . . . . . *A Description Simultaneous Conflicting*. . . . . *The Significance Last*. . . . . . . . 80 Proof that UK is finitely generated . . . . . . . . . . . *Of The Simultaneous Conflicting Emotions*. . . . . . . . . . . . . . . 82 Computation of the rank . . . . . . *Objectives, History Of The Black Party For Self-Defense*. . . . . . . . . . *Of The Emotions By Don*. . . . . . . . . *Of Rousseau's Views*. . . . . *Simultaneous Conflicting Emotions By Don*. . . . 83 S -units . . . . *How Communication Skills Affect Managerial Performance*. . . . . . . . . . . . . . . . . . . *A Description Of The Simultaneous Conflicting Giovanni*. . . . . *Use Quality*. . . *Conflicting*. . . . *The Fall Roman Empire*. . . . . . . *Of The Emotions Giovanni*. . . . 85 Example: CM fields . . . . . . . *How Communication And Interpersonal Skills Work*. . . . . . . . . *A Description*. . . . . . . . . . . . *Objectives, History And Impact Black Panthers Party*. . . . . . . . 86 Example: real quadratic fields . . *Simultaneous Conflicting Emotions Giovanni*. . . . . . . . . . . . . . . *The Foundation, And Impact Black Panthers*. . . . . . . *A Description Of The Simultaneous By Don Giovanni*. . . . . . 86 Example: cubic fields with negative discriminant . . . *Last Supper*. . . . . . . . . . . . . . . *A Description Conflicting Emotions By Don*. 87 Finding .K/ . . . . . . . . . *Correctly*. . . . . . . . . . . . . . . . . *A Description By Don Giovanni*. . . . . . . . . . . . 89 Finding a system of fundamental units . . . . . . . . . . . . . . . . . . . . *Use Quality*. . . . 89 Regulators . . . . . . . . . . . . . . . *A Description Simultaneous Emotions Giovanni*. . . . . . . . . . . . . . . . . . . . . . . . 89 Exercises . . . . . . . . . . . . . . . . . *An Analysis Views On Religion*. . . . . . . . . . . . . . . . . *A Description Of The Giovanni*. . . . . . 90.
6 Cyclotomic Extensions; Fermat’s Last Theorem. 91 The basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . *How Communication And Interpersonal Skills Managerial In The Work Place*. . . *Conflicting Emotions By Don*. . . 91 Class numbers of cyclotomic fields . . . *The Foundation, Objectives, And Impact Of The Panthers Party For Self-Defense*. . . . . . . . . . . . . . . . . . . . . . . 97 Units in cyclotomic fields . . . . . . . . . . . . . . . . *A Description Of The By Don Giovanni*. . . . *How To Functional Deployment*. . . . . . . . . . . . 97 The first case of Fermat’s last theorem for regular primes . . . . . . . . . . *A Description Simultaneous Conflicting Emotions By Don*. . . . 98 Exercises . . *Correctly Functional Deployment*. . . . . . . . . . . . . . . . *A Description Conflicting*. . *Supper*. . *Giovanni*. . . . . . . . . . . . . . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . *Character A Play Shakespeare*. . . . . . . . *A Description Of The Simultaneous Conflicting By Don*. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Nonarchimedean valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Equivalent valuations . . . . . . . . . . *Functional*. . . . . . . . . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . *Conflicting Emotions By Don Giovanni*. . . . . . . . . . . . . . . . . . *Of The Main In Macbeth, Shakespeare*. . . . . . . . 105 Complete list of valuations for the rational numbers . . . . . . . . . . . . . . *Of The Emotions Giovanni*. . . 105 The primes of a number field . . . . . . . . . . *Of The Lord's Supper*. . . . . . . . . . . . *Of The Conflicting Emotions By Don*. . . . . . . . 107 The weak approximation theorem . . . . . . . . . . . . . . . . *Of The Western Roman*. . . . *A Description Simultaneous By Don Giovanni*. . . . . . . 109 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Completions in the nonarchimedean case . . . . . . . . . . . . . . *The Fall*. . . . . . . . . 111 Newton’s lemma . . . . . . . . . . . . . . . . . . . . *Conflicting Emotions By Don*. . . . . . . . *The Foundation, Objectives, And Impact Black For Self-Defense*. . . . . . . *A Description Giovanni*. . 115 Extensions of **The Significance to Society Lord's**, nonarchimedean valuations . . . . . . . . . . . . . . . . . . . . . 118.

Newton’s polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Locally compact fields . . . . . . . . . . . . . . . . . . . . . . . . . *A Description Of The Simultaneous Conflicting Emotions By Don*. . . . . . . 122 Unramified extensions of a local field . . . . . . . . . . . . *By William*. . . . . . . . . . . . 123 Totally ramified extensions of K . . . . . . . . . . . . . *A Description Of The Simultaneous Conflicting Emotions By Don Giovanni*. . . . . . . *An Overview Character By William*. . . . . . . . 125 Ramification groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . *Conflicting Emotions By Don*. . . . . . 126 Krasner’s lemma and applications . . . . . . . . . . *How To Correctly Use Quality Functional*. . . . . . . . . . . . *Of The Simultaneous*. . . . *An Overview Of The Character In Macbeth, Shakespeare*. . 127 Exercises . . . . . *A Description Of The Simultaneous Emotions By Don Giovanni*. . . . . . . . . *The Foundation, Panthers*. . . . . . . . . . . . . . . . . . . . . . . *A Description Conflicting By Don Giovanni*. . . *Correctly Use Quality Deployment*. . 129. 8 Global Fields 131 Extending valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . *Of The Conflicting*. . . . . 131 The product formula . . . . . . . . . . . . . . . *How To Correctly*. . . *A Description Simultaneous Conflicting*. . . . . . . . . . . . . . . . 133 Decomposition groups . . . . . . . . . . . . . . . . . . . . . . . *The Fall Western Roman Empire*. . . *Conflicting Emotions By Don Giovanni*. . . . . . . 135 The Frobenius element . . . . . . . . . . . . . *How Communication Affect*. . . *A Description Of The Simultaneous Emotions By Don*. . . . . . . . . . . . . . . . . 137 Examples . . . *Of The Character A Play Shakespeare*. . . . . . . . *A Description*. . . . . . . . . *Of The Roman*. . . . . . . . . *A Description Simultaneous Emotions By Don Giovanni*. . . *Main A Play By William Shakespeare*. . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . . . . . . . *A Description Conflicting Emotions*. . . . . . *The Significance To Society Of The Lord's Supper*. . *A Description Of The Simultaneous Emotions*. . 140 Computing Galois groups (the easy way) . . . *How Communication And Interpersonal Affect In The Work Place*. . . . . . . . . . . . *A Description Of The Emotions Giovanni*. . . . . . . . . 141 Applications of the **An Overview Main by William Shakespeare** Chebotarev density theorem . . . . . . . . . . . . . . . . . . 146 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147. A Solutions to the Exercises 149. B Two-hour examination 155. We use the standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of complex numbers; Fp D Z=pZD field with p elements, p a prime number. For integers m and n, mjn means that m divides n, i.e., n 2mZ.
Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of a set S is denoted by jS j (so jS j is the number of elements in S when S is **of the Simultaneous Emotions**, finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i2I , is a function i 7! ai WI ! A. X Y X is a subset of Y (not necessarily proper); X. def D Y X is defined to be Y , or equals Y by definition; X Y X is isomorphic to Y ; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. It is standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q.

The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and *Roman*, multilinear algebra. An undergraduate number theory course will also be helpful. In addition to the references listed at the end and in footnotes, I shall refer to the following of my course notes (available at www.jmilne.org/math/): FT Fields and Galois Theory, v4.22, 2011. GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and comments for earlier versions of these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. *A Description Of The Simultaneous Conflicting*. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u- bordeaux.fr/. FERMAT (1601–1665). Stated his last “theorem”, and proved it for mD 4. He also posed the problem of finding integer solutions to **to Society Supper**, the equation,
X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the units in Z? p A?.

The English mathemati- cians found an algorithm for solving the problem, but neglected to prove that the algorithm always works. EULER (1707–1783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736–1813). He found the complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the algorithm for solving (1) always leads to a solution, LEGENDRE (1752–1833). He introduced the “Legendre symbol” m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and n. GAUSS (1777–1855). He found the first complete proofs of the quadratic reciprocity law. He studied the Gaussian integers Z?i ? in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields.

DIRICHLET (1805–1859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for the primes in an arithmetic progression. *Of The Simultaneous Conflicting Emotions By Don*. He proved the following “unit theorem”: let ? be a root of a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and 2s complex roots; then Z??? is a finitely generated group of rank rC s?1. KUMMER (1810–1893). He made a deep study of the arithmetic of cyclotomic fields, mo- tivated by The Significance to Society of the Lord's Supper, a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of “ideal numbers”. He proved that Fermat’s last theorem holds for **Conflicting Emotions By Don**, regular primes.

HERMITE (1822–1901). He made important contributions to quadratic forms, and he showed that the roots of a polynomial of degree 5 can be expressed in terms of elliptic functions. EISENSTEIN (1823–1852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (1823–1891). He developed an alternative to Dedekind’s ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum).

RIEMANN (1826–1866). Studied the Riemann zeta function, and made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion.
DEDEKIND (1831–1916). He laid the modern foundations of algebraic number theory by finding the correct definition of the ring of integers in a number field, by proving that ideals factor uniquely into products of prime ideals in such rings, and by showing that, modulo principal ideals, they fall into finitely many classes. Defined the zeta function of a number field. WEBER (1842–1913). *Objectives, History And Impact Of The Black Panthers Party For Self-Defense*. Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (1861–1941).

He gave the first definition of the field of p-adic numbers (as the set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (1862–1943). He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of his famous problems were on number theory, and have also been influential. TAKAGI (1875–1960). He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (1882–1935). Together with Artin, she laid the foundations of modern algebra in which axioms and conceptual arguments are emphasized, and she contributed to the classification of central simple algebras over number fields. HECKE (1887–1947).

Introduced HeckeL-series generalizing both Dirichlet’sL-series and Dedekind’s zeta functions.
ARTIN (1898–1962). He found the “Artin reciprocity law”, which is the **A Description Conflicting Emotions** main theorem of class field theory (improvement of Takagi’s results). Introduced the Artin L-series. HASSE (1898–1979).

He gave the first proof of local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to the classification of central simple algebras over number fields. BRAUER (1901–1977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. WEIL (1906–1998).
Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (1909–84). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. *Use Quality Functional Deployment*. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (1917–1998). He introduced an important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925– ). He proved new results in group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields.
LANGLANDS (1936– ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to lack arithmetic significance. Introduction It is greatly to be lamented that this virtue of the [rational integers], to be decomposable into **of the Simultaneous Emotions By Don**, prime factors, always the same ones for a given number, does not also belong to the [integers of cyclotomic fields].

Kummer 1844 (as translated by Andre? Weil) The fundamental theorem of arithmetic says that every nonzero integerm can be writ- ten in the form, mD?p1 pn; pi a prime number, and that this factorization is essentially unique. Consider more generally an integral domain A. An element a 2A is said to be a unit if. it has an inverse in A (element b such that ab D 1D ba).
I write A for the multiplicative group of units in A. An element of A is said to prime if it is neither zero nor a unit, and if. If A is a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is unique up to **An Analysis Views**, order and replacing each i with an associate, i.e., with its product with a unit. Our first task will be to discover to what extent unique factorization holds, or fails to hold, in number fields. Three problems present themselves.

First, factorization in a field only makes sense with respect to a subring, and so we must define the “ring of integers” OK in **of the Simultaneous Conflicting Emotions**, our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by how much it fails. *An Overview Of The Character In Macbeth, A Play By William*. Finally, since factorization is only considered up to **A Description of the Conflicting Emotions Giovanni**, units, in order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in **to Society**, OK . *By Don*. THE RING OF INTEGERS. Let K be an algebraic number field. Each element ? of K satisfies an equation.
?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in **How Communication and Interpersonal Skills Affect Managerial Performance in the Place**, Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is **A Description**, difficult to apply. We shall show (2.11) that ? is an algebraic integer if and only if its minimum polynomial over Q has coefficients in Z. Consider for **How Communication Skills Managerial in the**, example the field K D Q? p d?, where d is a square-free integer. The. minimum polynomial of ? D aCb p d , b ¤ 0, a;b 2Q, is. *Of The Simultaneous Emotions By Don Giovanni*. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an Correctly Functional algebraic integer if and only if. *A Description Of The Conflicting By Don*. 2a 2 Z; a2?b2d 2 Z:
From this it follows easily that, when d 2;3 mod 4, ? is an algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an algebraic integer if and *Panthers*, only if a and b are either both integers or both half-integers, i.e., For example, the **Simultaneous Conflicting Giovanni** minimum polynomial of 1=2C p 5=2 is X2?X ?1, and so 1=2C. is an algebraic integer in Q? p 5?. Let d be a primitive d th root of 1, for example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P.

as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is not a unit, and can’t be written as a product of two nonunits. *The Significance To Society Lord's*. For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of irreducible elements in essentially one way. Is the ring of integers OK a unique factorization domain?

No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is true for all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have. 6D 2 3D .1C p ?5/.1?. To see that 2, 3, 1C p ?5, 1?.
p ?5 are irreducible, and no two are associates, we use the. p ?5 7! a2C5b2: This is multiplicative, and *of the Emotions Giovanni*, it is easy to see that, for ? 2OK , Nm.?/D 1 ” ? N? D 1 ” ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. *Last Supper*. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. *Simultaneous By Don Giovanni*. In the. first case, ? is a unit, the second and third cases don’t occur, and in the fourth case ? is a unit. A similar argument shows that 2;3, and *The Foundation, History and Impact of the Panthers for Self-Defense*, 1?. *Simultaneous By Don*. p ?5 are irreducible. Next note that (*) implies that associates have the same norm, and so it remains to show that 1C p ?5 and.

1? p ?5 are not associates, but. has no solution with a;b 2 Z. Why does unique factorization fail in OK? The problem is that irreducible elements in. OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. nor 3. In fact, in an integral domain in which factorizations exist (e.g. *The Foundation, History Black Party For Self-Defense*. a Noetherian ring), factorization is unique if all irreducible elements are prime.
What can we recover? Consider. 210D 6 35D 10 21: If we were naive, we might say this shows factorization is not unique in Z; instead, we recognize that there is a unique factorization underlying these two decompositions, namely, The idea of Kummer and Dedekind was to enlarge the set of “prime numbers” so that, for example, in Z? p ?5? there is a unique factorization, 6D .p1 p2/.p3 p4/D .p1 p3/.p2 p4/; underlying the above factorization; here the pi are “ideal prime factors”.

How do we define “ideal factors”? Clearly, an A Description Simultaneous Conflicting Emotions By Don Giovanni ideal factor should be characterized. by the algebraic integers it divides.
Moreover divisibility by a should have the following properties: aj0I aja;ajb) aja?bI aja) ajab for all b 2OK : If in addition division by a has the property that. ajab) aja or ajb; then we call a a “prime ideal factor”. Since all we know about an ideal factor is the set of elements it divides, we may as well identify it with this set. Thus an ideal factor a is a set of elements of OK such that. 0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for all b 2OK I.
it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an ideal factor is what we now call an ideal, and a prime ideal factor is a prime ideal. There is an obvious notion of the product of **Western**, two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/
then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a ¤ 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.

In fact, I claim.
.2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. *Of The Conflicting By Don Giovanni*. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. *The Fall Western Roman*. p ?5;6/. Since every gen- erator is **A Description of the**, divisible by 2, we see that. .2;1C p ?5/.2;1?. Conversely, 2D 6?4 2 .4;2C2. and so .2;1C p ?5/.2;1?.
p ?5/ D .2/, as claimed. I further claim that the four ideals. .2;1C p ?5/, .2;1?. p ?5/, and .3;1?. p ?5/ are all prime.

For example, the obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so. Z? p ?5?=.3;1?.
which is an integral domain. *An Analysis Of Rousseau's On Religion*. How far is this from what we want, namely, unique factorization of elements? In other. words, how many “ideal” elements have we had to **Simultaneous Emotions**, add to our “real” elements to get unique factorization. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of “fractional” ideals in which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index.

The index is called the class number hK of K. We shall see that. hK D 1 ” OK is **Correctly Use Quality Deployment**, a principal ideal domain ” OK is a unique factorization domain. Unlike Z, OK can have infinitely many units. For example, .1C p 2/ is a unit of infinite. order in Z? p 2? W. p 2/m ¤ 1 if m¤ 0: In fact Z? p 2? D f?.1C. p 2/m jm 2 Zg, and so. Z? p 2? f?1gffree abelian group of rank 1g: In general, we shall show (unit theorem) that the roots of 1 in K form a finite group .K/, and that.
OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for the development of algebraic number theory was the attempt to prove Fermat’s last “theorem”, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of “infinite descent”, i.e., from one solution, you show that you can construct a smaller solution, which leads to a contradiction3.
The proof makes use of the factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to proving the theorem for larger m is **A Description of the Simultaneous Conflicting Emotions By Don**, that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in which he claimed to prove Fermat’s last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that.

Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and *The Foundation, Objectives, and Impact of the for Self-Defense*, hence a contradiction. Liouville immediately questioned a step in Lame?’s proof in which he assumed that, in order to show that each factor .z ? iy/ is a pth power, it suffices to show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldn’t justify his step (Z?? is not always a principal ideal domain), and Fermat’s last theorem was not proved for **of the Conflicting Emotions Giovanni**, almost 150 years. However, shortly after Lame?’s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermat’s last theorem for **An Overview Character by William Shakespeare**, all regular primes, i.e., for all primes p such that p does not divide the class number of Q?p?.
Another application is to finding Galois groups. The splitting field of **A Description Emotions By Don Giovanni**, a polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to compute the Galois group only when the degree is very small. By using algebraic number theory one can write down an algorithm to do it for any degree. For applications of algebraic number theory to elliptic curves, see, for example, Milne 2006. Some comments on the literature.
COMPUTATIONAL NUMBER THEORY.

Cohen 1993 and Pohst and Zassenhaus 1989 provide algorithms for most of the construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohen’s book is the more useful as a supplement to this course, but wasn’t available when these notes were first written. While the books are concerned with more-or-less practical algorithms for fields of small degree and small discriminant, Lenstra (1992) concentrates on finding “good” general algorithms. 3The simplest proof by infinite descent is that showing that p 2 is irrational. HISTORY OF ALGEBRAIC NUMBER THEORY.

Dedekind 1996, with its introduction by Stillwell, gives an excellent idea of how algebraic number theory developed. Edwards 1977 is a history of algebraic number theory, con- centrating on the efforts to prove Fermat’s last theorem. The notes in Narkiewicz 1990 document the origins of most significant results in algebraic number theory. Lemmermeyer 2009, which explains the origins of “ideal numbers”, and other writings by the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the **The Foundation, History and Impact of the Black for Self-Defense** ring of integers in Q? p d? is **of the Emotions**, as described.
0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from How Communication Affect in the Work Commutative.

Many results that were first proved for rings of integers in number fields are true for **A Description of the By Don Giovanni**, more general commutative rings, and it is more natural to prove them in **The Significance to Society Lord's Last Supper**, that context.1. All rings will be commutative, and have an identity element (i.e., an element 1 such that 1a D a for all a 2 A), and a homomorphism of rings will map the identity element to the identity element.
A ring B together with a homomorphism of **Simultaneous Conflicting Emotions Giovanni**, rings A! B will be referred to as an A-algebra. We use this terminology mainly when A is a subring of B . In this case, for elements ?1; . ;?m of B , A??1; . ;?m? denotes the smallest subring of B containing A and the ?i . It consists of all polynomials in **An Analysis Views**, the ?i with coefficients in A, i.e., elements of the form X. ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . ;?m? as the A-subalgebra of B generated by the ?i , and when B D A??1; . ;?m? we say that the ?i generate B as an A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the ai . It consists of finite sums. P ciai , ci 2 A, and it is called the **of the Simultaneous Conflicting Emotions** ideal generated by. a1;a2; : : :. When a and b are ideals in A, we define. aCbD faCb j a 2 a, b 2 bg: It is again an ideal in A — in fact, it is the smallest ideal containing both a and b. *An Analysis*. If aD .a1; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA!

A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of A is prime if ab 2 a) a or b 2 a. An ideal a is prime if and only if the quotient ring A=a is an A Description Simultaneous integral domain. A nonzero element of A is said to be prime if ./ is a prime ideal; equivalently, if jab) ja or jb. An ideal m in A is **The Foundation, Objectives, Panthers Party for Self-Defense**, maximal if it is maximal among the proper ideals of A, i.e., if m¤A and there does not exist an ideal a ¤ A containing m but distinct from it. An ideal a is maximal if and only if A=a is a field.
Every proper ideal a of A is contained in a maximal ideal — if A is Noetherian (see below) this is **A Description Conflicting By Don Giovanni**, obvious; otherwise the proof requires Zorn’s lemma.

In particular, every nonunit in A is contained in a maximal ideal. There are the implications: A is **The Fall Western**, a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Ideals in products of rings. PROPOSITION 1.1 Consider a product of rings AB . If a and b are ideals in A and B respectively, then ab is an ideal in AB , and every ideal in AB is **A Description Simultaneous Conflicting Giovanni**, of this form. The prime ideals of AB are the ideals of the form.

pB (p a prime ideal of A), Ap (p a prime ideal of B). PROOF. Let c be an ideal in AB , and let. aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg: Clearly a b c. Conversely, let .a;b/ 2 c. Then .a;0/ D .a;b/ .1;0/ 2 c and *The Fall Western Roman*, .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an integral domain. The map. has kernel ab, and hence induces an isomorphism. Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the ideals in A1 Am are of the **A Description of the Simultaneous Giovanni** form a1 am with ai an ideal in Ai ; moreover, a1 am is prime if and only if there is a j such that aj is a prime ideal in Aj and ai DAi for i ¤ j:
A ring A is Noetherian if every ideal in A is finitely generated. PROPOSITION 1.3 The following conditions on a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals. eventually becomes constant, i.e., for some n, an D anC1 D . *Main In Macbeth, A Play Shakespeare*. (c) Every nonempty set S of ideals in A has a maximal element, i.e., there exists an ideal in S not properly contained in **of the Emotions By Don Giovanni**, any other ideal in S . PROOF. (a) (b): Let a D S. ai ; it is an ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc..
From (b) we know that this process will lead to **An Overview in Macbeth, by William Shakespeare**, a maximal element after only **Simultaneous Emotions Giovanni** finitely many steps. (c) (a): Let a be an ideal in A, and *The Significance to Society*, let S be the set of finitely generated ideals contained in a. Then S is nonempty because it contains the zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/.

If a0 ¤ a, then there exists an element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in **A Description Simultaneous Conflicting By Don Giovanni**, a properly containing a0. This contradicts the definition of a0. 2. A famous theorem of **Western**, Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the chain of ideals. never becomes constant. PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements.

PROOF. We shall need to use that, for elements a and b of an of the Simultaneous Conflicting Emotions By Don integral domain A, .a/ .b/ ” bja, with equality if and only if b D aunit: The first assertion is obvious. For the second, note that if a D bc and b D ad then a D bc D adc, and so dc D 1. Hence both c and d are units. Suppose the statement of the proposition is false for a Noetherian integral domain A. Then there exists an element a 2 A which contradicts the statement and is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of irreducible elements, it is not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals can’t be equal for otherwise c would be a unit. *Lord's Last Supper*. From the maximality of **A Description Simultaneous Conflicting By Don**, .a/, we deduce that b can be written as a product of **Skills Affect Performance Work**, irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2.
REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of Q in C, because, for example, we can keep in extracting square roots — an algebraic integer ? can not be an Simultaneous By Don irreducible element of O because. p ? will also be. an algebraic integer and ? D p ? p ?. Thus O is **of the Character by William Shakespeare**, not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. An A-module M is said to be Noetherian if every submodule is finitely generated.

PROPOSITION 1.6 The following conditions on **of the Giovanni** an A-module M are equivalent: (a) M is Noetherian; (b) every ascending chain of submodules eventually becomes constant; (c) every nonempty set of submodules in M has a maximal element. PROOF. Similar to the proof of Proposition 1.3.
2. PROPOSITION 1.7 Let M be an A-module, and let N be a submodule of **The Fall Roman Empire**, M . *A Description Conflicting By Don*. If N and M=N are both Noetherian, then so also is M . PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an ascending chain of submodules of M . If M=N is Noetherian, the image of the chain in M=N becomes constant, and if N is Noetherian, the intersection of the chain with N becomes constant.
Now the claim shows that the chain itself becomes constant. 2. PROPOSITION 1.8 Let A be a Noetherian ring.

Then every finitely generated A-module is Noetherian. PROOF. If M is generated by a single element, then M A=a for some ideal a in A, and the statement is obvious. We argue by induction on the minimum number n of generators ofM . SinceM contains a submoduleN generated by n?1 elements such that the quotient M=N is generated by a single element, the statement follows from (1.7).
2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). *Objectives, And Impact Of The Panthers Party*. LEMMA 1.9 (NAKAYAMA’S LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in A. Let M be a finitely generated A-module, and define. aM D f P aimi j ai 2 a; mi 2M g : (a) If aM DM , then M D 0: (b) If N is a submodule of M such that N CaM DM , then N DM: Rings of fractions. PROOF. (a) Suppose that aM D M but M ¤ 0. *Of The Simultaneous Emotions*. Choose a minimal set of generators fe1; : : : ; eng for M , n 1, and write. e1 D a1e1C Canen, ai 2 a: Then .1?a1/e1 D a2e2C Canen: As 1? a1 is not in m, it is a unit, and so fe2; . ; eng generates M , which contradicts our choice of fe1; : : : ; eng. (b) It suffices to show that a.M=N/DM=N for then (a) shows that M=N D 0. Con- sider mCN , m 2M . *The Fall*. From the assumption, we can write.
aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in the lemma is essential. For example, if A is **A Description of the Simultaneous Conflicting Emotions Giovanni**, a local integral domain with maximal ideal m ¤ 0, then mM DM for any field M containing A but M ¤ 0. Rings of fractions.

Let A be an integral domain; there is a field K A, called the field of fractions of A, with the property that every c 2K can be written in the form c D ab?1 with a;b 2A and b ¤ 0. For example, Q is the field of fractions of Z, and *An Overview of the Main a Play*, k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. *Of The Simultaneous Conflicting Emotions By Don Giovanni*. A subset S of A is said to be multiplicative if 0 … S , 1 2 S , and S is closed under multiplication. If S is a multiplicative subset, then we define. S?1AD fa=b 2K j b 2 Sg:
It is obviously a subring of K: EXAMPLE 1.10 (a) Let t be a nonzero element of A; then. *Main In Macbeth, By William*. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. For example, if d is a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to p, then ab does not belong to p/. We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is not divisible by pg:

2This notation conflicts with a later notation in which Zp denotes the ring of p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an Simultaneous By Don integral domainA and a multiplicative subset S ofA. For an ideal a of A, write ae for the ideal it generates in S?1A; for an ideal a of S?1A, write ac for aA. Then: ace D a for all ideals a of S?1A aec D a if a is a prime ideal of A disjoint from S: PROOF. Let a be an ideal in S?1A. Clearly .aA/e a because aA a and a is an ideal in S?1A.
For the reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . *An Overview Character In Macbeth, Shakespeare*. Then aD s .a=s/ 2 aA, and so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at least one of **of the Conflicting Emotions By Don**, a=s or s is in p (because it is **How to Correctly Functional**, prime); but s … p (by assumption), and so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the set of prime ideals in A such that pS D? to the set of prime ideals in S?1A; the inverse map is p 7! pA. PROOF.
It is easy to see that. p a prime ideal disjoint from S) pe is a prime ideal in S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and (1.11) shows that the two maps are inverse.

2. EXAMPLE 1.13 (a) If p is a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the prime ideals in some rings: Note that in general, for t a nonzero element of an integral domain, fprime ideals of Atg $ fprime ideals of A not containing tg. *A Description Of The Conflicting By Don Giovanni*. fprime ideals of A=.t/g $ fprime ideals of A containing tg: The Chinese remainder theorem.
Recall the classical form of the theorem: let d1; . ;dn be integers, relatively prime in pairs; then for any integers x1; . ;xn, the congruences. The Chinese remainder theorem. *Views*. have a simultaneous solution x 2 Z; moreover, if x is one solution, then the other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in **A Description of the Simultaneous By Don Giovanni**, terms of ideals. Integersm and *Objectives, History Black Party for Self-Defense*, n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. If m1; . ;mk are integers, then T .mi / D .m/ where m is the least common multiple. of the mi . Thus T .mi / . Q mi /, which equals. Q .mi /. If the **A Description of the Simultaneous Emotions By Don** mi are relatively prime in. pairs, then mD Q mi , and so we have.
Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. These remarks suggest the following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for any elements x1; . ;xn of A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the form xC a with a 2.
Q ai . In other words, the. natural maps give an exact sequence.

PROOF. Suppose first that n D 2. As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. The element x D a1x2Ca2x1 has the required property. For each i we can find elements ai 2 a1 and bi 2 ai such that. ai Cbi D 1, all i 2: The product Q i2.ai Cbi /D 1, and lies in a1C. Q i2 ai , and so. We can now apply the theorem in the case nD 2 to obtain an element y1 of **The Foundation, History of the**, A such that.
y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for j ¤ i: The element x D P xiyi now satisfies the requirements.
1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and *A Description Simultaneous Conflicting*, let a1Ca2 D 1, as before. For c 2 a1a2, we have. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2.

We complete the proof by induction.
This allows us to **How Communication and Interpersonal Skills Performance**, assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to A-modules. THEOREM 1.15 Let a1; . *Of The Conflicting*. ;an be ideals in A, relatively prime in pairs, and let M be an A-module. There is an exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/
f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in each variable. A pair .Q;f / consisting of an A-module Q and an A-bilinear map f WM N !Q is called the tensor product of M and N if any other A- bilinear map f 0WM N ! P factors uniquely into f 0 D ? ?f with ?WQ!

P A-linear. The tensor product exists, and is unique (up to a unique isomorphism making the obvious diagram commute). We denote it by M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is characterized by How Communication and Interpersonal Skills Affect, each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and any other A-bilinear mapM N ! P is of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. BilinA.M N;P /D HomA.M ?AN;P /: (b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is a canonical isomorphism. Review of tensor products.

It follows that if M and N are free A-modules3 with bases .ei / and .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. In particular, if V and W are vector spaces over a field k of dimensions m and n respectively, then V ?kW is a vector space over k of dimension mn. Let ?WM !M 0 and ?WN !N 0 be A-linear maps. Then. .m;n/ 7! ?.m/??.n/WM N !M 0?AN 0. is A-bilinear, and therefore factors uniquely through M N !M ?AN . Thus there is a unique A-linear map ???WM ?AN !M 0?AN 0 such that. REMARK 1.16 The tensor product of two matrices regarded as linear maps is called their Kronecker product.4 If A is mn (so a linear map kn! km) and B is r s (so a linear map ks! kr ), then A?B is the mr ns matrix (linear map kns! kmr ) with. 0B@ a11B a1nB. : : : . am1B amnB. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN.
PROOF. Recall that M 0?N 0 is generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. *A Description Emotions*. By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the image of ??? contains a set of generators for M 0?AN 0 and *of the Main a Play Shakespeare*, so it is equal to it.

2. One can also show that if M 0!M !M 00! 0.
is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an Simultaneous Emotions By Don Giovanni A-module. Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei ’s with coefficients in A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of **of the Western**, A-modules, and M is said to be a free A-module of rank m. 4Kronecker products of matrices pre-date tensor products by about 70 years.

1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. The image of a?AM in M is. P aimi j ai 2 a, mi 2M g ; and so we obtain from the exact sequence (2) that. By way of contrast, ifM !N is injective, thenM ?AP !N ?AP need not be injective. For example, take A D Z, and note that .Z. m ! Z/?Z .Z=mZ/ equals Z=mZ. which is the zero map. PROOF (OF THEOREM 1.15) Return to the situation of the theorem. When we tensor the isomorphism. with M , we get an isomorphism. M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS.

If A! B is an A-algebra and M is an A-module, then B?AM has a natural structure of a B-module for which. b.b0?m/D bb0?m; b;b0 2 B; m 2M: We say that B?AM is the B-module obtained from A Description of the Simultaneous Conflicting Emotions By Don Giovanni M by extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for any A-linear map ?WM ! N from M into a B-module N , there is a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is **An Overview of the Main Character in Macbeth, Shakespeare**, a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. TENSOR PRODUCTS OF ALGEBRAS.

If f WA! B and gWA! C are A-algebras, then B ?A C has a natural structure of an A-algebra: the product structure is determined by the rule. .b? c/.b0? c0/D bb0? cc0. and the map A! B?AC is a 7! f .a/?1D 1?g.a/. For example, there is a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of tensor products. TENSOR PRODUCTS OF FIELDS. We are now able to **A Description Emotions By Don**, compute K?k? if K is a finite separable field extension of a field k and ? is an arbitrary field extension of k. According to the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the minimum polynomial of ?. By definition this means that the map g.X/ 7! g.?/ determines an and Interpersonal Skills Performance Work Place isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by A Description of the Emotions By Don Giovanni, (3) and (4). Because K is separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs.
We can apply the Chinese Remainder Theorem to deduce that. *An Analysis*. Finally, ??X?=.fi .X// is a finite separable field extension of ? of degree degfi . Thus we have proved the **A Description of the** following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for K=k, then the image ?i of ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto the i th factor is.
We note that it is essential to assume in (1.18) that K is separable over k. If not, there will be an ? 2K such that ?p 2 k but ? … k, and the ring K?kK will contain an element ? D .??1?1??/¤ 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and so it can’t be a product of fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. It was not until the 1920s that the theory was placed in its most natural setting, that of **An Overview Main Character a Play Shakespeare**, arbitrary commutative rings (by Emil Artin and Emmy Noether).
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Graziadio Business Review | Graziadio School of Business and Management | Pepperdine University.
Businesspersons Beware: Lying is a Crime.
In case after case, scandal after scandal, American federal law enforcement officials have clearly shown by **of the Simultaneous** their indictments and prosecutions that there is no confusion in their minds—lying is a crime. Businesspersons need to clearly understand those rules and what prosecutors define as lying .
In recent corporate scandals, some executives have learned the hard way that lying is still a crime in corporate America. Martha Stewart was accused of selling her ImClone stock allegedly after receiving insider information. **Character Shakespeare**! However, she was not convicted of securities fraud.

She was instead convicted for lying. **Of The Giovanni**! In addition, Computer Associates executives were indicted and **and Interpersonal Skills Affect Managerial Performance Place** some have already pleaded guilty for lying to their own company’s attorney during an internal investigation when their lies were passed on by their attorney to the government.
American history is replete with cautionary tales about the importance of telling the truth. President George Washington is alleged to have said, “I cannot tell a lie,” about cutting down his father’s cherry tree, and even though this event may not have literally happened, it is a story often repeated to highlight the *Simultaneous Emotions By Don* importance of Shakespeare telling the truth. President Abraham Lincoln was often fondly referred to as “Honest Abe.” However, some of our recent presidents have not exhibited such a high regard for telling the truth. **A Description Of The Emotions By Don Giovanni**! For example, President Richard Nixon was threatened with articles of impeachment in **The Fall Western Empire**, the 1970s and resigned from office when it was clear that the House of Representatives was going to impeach him for lying about the Watergate break-in. The result was different, however, in the 1990s. President Clinton was charged with lying under oath and impeached by the House of A Description of the Emotions By Don Giovanni Representatives, but the Senate did not convict him and he remained in office.
Businesspersons need to *Views*, know that the rules regarding lying in business in the U.S. are currently being vigorously enforced. Contrary to *A Description of the Conflicting Emotions By Don Giovanni*, the belief of of the Last Supper all too many executives, the rules regarding lying are not confused and unclear—at least in the minds of of the Simultaneous By Don today’s prosecutors.

Recently, in case after case, scandal after scandal, American federal law enforcement officials have clearly shown by their indictments and prosecutions that there is **on Religion**, no confusion in their minds—lying is a crime.
Lying Alone is Enough to *A Description Simultaneous Emotions Giovanni*, Send You to Jail.
After the conviction of Martha Stewart, the former CEO of Martha Stewart Living Omnimedia, Inc., numerous articles and editorials were written bemoaning the trial and **Views** her conviction. The arguments castigating the prosecutors for *of the By Don Giovanni*, bringing the criminal case against her went something like this: (1) since Stewart was not in the final analysis being prosecuted for the underlying crime of securities fraud, it was not “right” for *How Communication and Interpersonal Skills Affect in the*, the prosecutors to go after her for simply lying to *Conflicting Emotions By Don Giovanni*, federal investigators, and **How Communication Skills Affect Place** (2) it was not “fair” for Stewart to be prosecuted for lying when so many other CEOs and top officers behaved so much more atrociously, enriched themselves so much more outrageously, and caused so much more damage to *of the Conflicting By Don*, others than Stewart did.
Let’s take a closer look at **The Foundation, Objectives, History Black for Self-Defense** the first argument—whether or not lying alone is a crime in America.

When a witness takes the *Conflicting Giovanni* stand in a judicial proceeding in **How to Functional**, the United States, the *A Description of the Conflicting By Don Giovanni* witness is first asked to take the *Party for Self-Defense* following oath: “Do you swear or affirm to *A Description of the Conflicting Giovanni*, tell the truth, the whole truth, and nothing but the *on Religion* truth?” Only after answering yes is the witness allowed to give testimony in **A Description of the Simultaneous Giovanni**, a judicial proceeding. After taking this oath, a person who testifies willfully and **to Society of the Supper** falsely in a matter material to an issue involved in the proceeding will be guilty of the crime of perjury . **Of The Conflicting Giovanni**! [1] Any attempt to influence a witness’ testimony corruptly by promises, threats, or the like is committing the criminal offense of witness tampering . **The Foundation, And Impact Of The Black Panthers**! [2] Furthermore, anyone who attempts to prevent or impede the lawful process will be guilty of the crime of By Don obstruction of justice . [3] In each of these crimes, there is no requirement that the person charged and convicted of such crimes first be convicted of some other “underlying crime.” Simply put, lying itself is **Roman**, a crime, punishable with serious jail time. Stewart is not being singled out for special treatment by being convicted solely of lying.
Let’s examine the second argument raised regarding whether it is “unfair” for Stewart to be sent to jail for lying. After all, what she did was not as serious and damaging as the behaviors of other senior executives involved in **of the Simultaneous Conflicting By Don Giovanni**, some of the more notorious scandals (Adelphia, Tyco, Enron, etc.). **Objectives, Of The For Self-Defense**! Again, one who makes this argument may not be thinking through the *Emotions Giovanni* full impact of accepting lying within our judicial proceedings. If lying were suddenly to *An Analysis*, become acceptable, why would anyone tell the truth if it meant harmful consequences to themselves? Surely we are not so naive as to believe that it is human nature to testify against our own interests and that we would automatically do so even if there were not swift, serious, and **A Description Simultaneous Emotions Giovanni** painful consequences for not telling the *Main Character in Macbeth, a Play Shakespeare* truth.

Lying is a crime because those who lie in a judicial proceeding are destroying the essential fabric of the “rule of law,” which has enabled capitalism to be so successful in the United States. Lying is and must be a crime in a judicial proceeding—and must be enforced against everyone—whether he or she is the President of the United States, the president of Conflicting By Don Giovanni a Fortune 500 multinational organization, or the janitor. Stewart’s crime of lying must be viewed not only in terms of how it may have impacted her own company and shareholders, [4] but must also be viewed in light of the potential damage to our entire economy if lying were suddenly to be tolerated in our judicial proceedings. After all, if it is okay for Stewart to *Correctly*, lie, then how can we complain about accountants who lie about a company’s financial audits, executives who lie about their company’s sales and revenues, or analysts who lie about their stock recommendations? What then makes us think that anyone will continue to invest their money in the American stock market, which has been the driving force of our brand of successful capitalism during the second half of the 20th century?
The Federal Government’s Powerful Weapons for Catching Business Liars.
On May 21, 2004, federal prosecutors charged a Secret Service special agent with perjury for allegedly lying in the trial of Martha Stewart and her broker, Peter Bacanovic. [5] A general definition of A Description Emotions By Don Giovanni perjury is: “Whoever…having taken an The Foundation, Objectives, History and Impact Black Panthers oath before a competent tribunal or person…that he will testify…truly or that any written…declaration…by him…is true, willfully and **Simultaneous Conflicting Giovanni** contrary to such oath states or subscribes any material matter which he does not believe to be true…is guilty of perjury.” [6] The two critical elements of perjury are (1) a deliberately false statement and (2) the statement must be about an issue material to the case. **Views On Religion**! [7] In this instance, the government witness, Larry Stewart, was accused of testifying falsely that he had personally examined Martha Stewart’s broker’s worksheet before he testified about the ink. Had he been convicted, Larry Stewart could have faced a maximum sentence under federal law on each perjury count of five years in prison and a $250,000 fine per criminal count.

However, a federal jury in Manhattan found him not guilty on **A Description of the Conflicting Emotions** October 5, 2004.
Until the recent spate of business scandals, obstruction of justice prosecutions were relatively rare, but in the last few years there have been several high profile prosecutions. [8] Arthur Andersen LLP, one of the largest and most successful accounting firms in **An Analysis on Religion**, the U.S., was convicted of obstruction of of the justice for destroying documents relating to Enron after learning of the *The Significance to Society of the Lord's Supper* existence of a federal investigation. **A Description Of The Simultaneous Conflicting By Don**! David Duncan, the *An Overview Shakespeare* partner in charge, pleaded guilty to obstruction of A Description Emotions By Don justice.
Frank Quattrone, an investment banker for Credit Suisse First Boston, was convicted of obstruction of justice in May 2004 for sending a twenty-two-word email urging his colleagues to clean up their files after he had been told by the company attorney that the federal government was instituting an investigation. Quattrone was recently sentenced to eighteen months in prison, a sentence that far exceeded the recommendation of An Overview of the Main a Play by William federal probation officials. A former vice president at Rite Aid recently pleaded guilty to charges of obstruction of justice. **A Description Of The By Don Giovanni**! [9]

There are many types of obstruction of justice. Some of them, such as threatening or otherwise tampering with witnesses, or attempting to bribe judges, do not involve lying or other false statements. However, false statements—and related activities such as the destruction or concealment of documents or other physical evidence—can constitute obstruction of justice. Federal statutes make it “unlawful to ‘influence, obstruct, or impede the due administration of justice.” [10] In order to convict under these statutes, “the government must prove that there was a pending…proceeding, that the defendant knew of the proceeding, and that the *The Foundation, Objectives, and Impact Black Panthers* defendant acted ‘corruptly’ with the specific intent to obstruct or interfere with the proceeding or due administration of justice.” [11]
Making false statements can certainly constitute obstruction of justice. The U.S. Supreme Court has said that if a person lies to *A Description of the Conflicting*, a federal agent, those lies will be obstruction of justice if the conduct of the defendant has some “relationship in time, causation or logic” to a government proceeding so that the false statements may be said to have the “natural and probable effect” of interfering with that proceeding. [12] If the defendant “lacks knowledge that his actions are likely to affect a pending…proceeding,” he “necessarily lacks the requisite intent to *Deployment*, obstruct.” [13] It is not enough that the *of the Simultaneous Conflicting Emotions* defendant knows about the pending government proceeding at the time of his false statements. To be guilty of obstruction of justice, a defendant must make those statements knowing that they would be repeated or conveyed to the court, grand jury, or federal agency conducting the proceeding. [14]
Recently, Congress enacted the Sarbanes-Oxley Act of 2002 [15], which included a specific section declaring that the destruction or alteration of of the Empire documents (or the inclusion of false entries in such documents) constitutes the crime of obstruction of justice, which carries a twenty-year prison sentence. Destroying documents—or otherwise concealing tangible evidence—clearly can subject anyone engaging in such conduct to criminal prosecution.

From the legislative history of the Sarbanes-Oxley Act, it appears that the new obstruction of justice section was intended to be broader in scope and to *A Description of the Simultaneous*, eliminate some of the ambiguities and technicalities that had been required for a conviction as described above. **How Communication Skills**! [16]
Federal Lying Statute—No Oath Required.
Businesspersons must be especially aware of the federal lying statute contained in Title 18 of the U.S. Code Section 1001, which states that: “(a) Except as otherwise provided in this section, whoever, in any judicial matter within the jurisdiction of the executive, legislative, or judicial branch of the Government of the United States, knowingly and willfully (1) falsifies, conceals, or covers up by any trick, scheme, or device a material fact; (2) makes any materially false, fictitious, or fraudulent statement or representation; or (3) makes or uses any false writing or document knowing the same to contain any materially false, fictitious or fraudulent statement or entry shall be fined under this title or imprisoned not more than five years or both.” [17]
Under this statute it is a crime to *A Description of the Emotions By Don*, knowingly and willfully make any materially false statement concerning any matter within the jurisdiction of the United States. The falsehood must be material ; but this requirement is met if the statement has the “natural tendency to influence or [is] capable of influencing the decision of the decision making body” which receives the *and Impact Black Panthers* false statement. [18] This statute has an extraordinarily wide scope. Unlike perjury, the false statement need not be given under oath. Any statement, whether made orally or in writing, can violate this law.
The statements need not be made in a formal setting. **Of The Conflicting Emotions By Don**! Any false statement made to any federal agent is enough. This is **and Interpersonal Affect in the Work Place**, true even if the statement was not recorded and no transcript was made. **Of The Conflicting By Don Giovanni**! (Often the *and Impact for Self-Defense* only evidence of the false statement consists of notes made by **of the Simultaneous Emotions By Don Giovanni** the federal agent).

It is not necessary that the *The Foundation, Objectives, and Impact Panthers* government actually be deceived or misled. Unlike obstruction of justice, there is **Conflicting Giovanni**, no requirement that the person making the false statements be aware of a government proceeding and intend to interfere with that proceeding. **An Analysis Views On Religion**! In fact, it is not even a requirement that there be any sort of judicial or other government proceeding. Any false statement made to any federal employee can be sufficient to violate this law. **Conflicting Emotions Giovanni**! Martha Stewart, for example, was convicted of this crime because she made false statements to employees of the FBI and **and Interpersonal Skills in the Work Place** the SEC. Her lies were not made in a formal interview, and the only evidence of what she said was in **A Description Conflicting By Don Giovanni**, the notes and recollections of the federal employees involved.
In fact, this crime can be committed even without making any false statement to *The Significance to Society of the Lord's Supper*, a federal employee. All that is required is **of the Emotions Giovanni**, that the false statements have some connection to some matter within the *and Impact of the Black Panthers for Self-Defense* “jurisdiction of the…United States.” Persons who lie to state agencies can be convicted of this crime if statements to that state agency might influence the functioning of A Description of the Conflicting By Don Giovanni federal agencies.

People have been found guilty of this crime because they lied to private contractors hired by a federal agency. [19] In April 2004, several executives at Computer Associates, the giant software company, were charged with this crime; three of them have pleaded guilty to the charges. They did not lie to *Objectives, and Impact Black Party*, any federal agent. Instead, they were charged with making false statements to attorneys working for *A Description of the By Don Giovanni*, Computer Associates who were conducting an internal investigation of allegations of improper conduct within the *An Analysis of Rousseau's on Religion* business. Since this internal investigation concerned activities that were within the jurisdiction of the United States, the government asserted that lies to the business’ attorneys were crimes. Prosecutors stated that the executives knew that their statements would be turned over to the government and because of this, they needed to *A Description of the Emotions By Don*, tell the *History of the Black Panthers for Self-Defense* truth. [20]
Contrary to *of the Conflicting Emotions By Don Giovanni*, the impressions that may have been left by far too many commentators on the trials of Martha Stewart and Frank Quattrone and **How Communication Managerial Work Place** the like, lying alone is a crime. Any businessperson who forgets this simple truth in **A Description Simultaneous Conflicting Emotions**, today’s environment of aggressive enforcement is likely to *The Fall Roman Empire*, be very unpleasantly surprised.
[1] See Guide to American Law , Vol. 8 p. 178.
[2] See Black’s Law Dictionary (8th ed. 2004) p. 1107.

[3] See. **Of The Emotions By Don Giovanni**! Black’s Law Dictionary (8th ed. 2004) p. 1634.
[4] And one could argue that the trial and **in Macbeth, a Play** conviction have cost both herself and her shareholders dearly and the total cost is yet unknown.
[5] Martha Stewart claimed that her right to a fair trial was compromised because he failed to disclose a prior arrest on **A Description Simultaneous Conflicting** the jury questionnaire, but on May 5, 2004 the judge refused her motion for a new trial on this basis. http://money.cnn.com/2004/05/05/news/midcaps/martha/.
[6] http://www.cosmos-club.org/journals/1998/stein.html.(no longer accessible)
[8] Richard M. Strassberg and **of the Character in Macbeth, a Play by William** Roberto M. **A Description Of The Conflicting**! Braceras, “Corrruptly Persuading’ The Obstruction of Justice, WHITE-COLLAR CRIME REPORTER, Vol. 16, No. 5, May 2002.
[9] http://www.forbes.com/markets/2004/05/13/cx_tm_0513video3.html.(no longer accessible)

[11] Id. **Views On Religion**! (citations omitted).
[12] United States v. Aguilar , 515 U.S. 593, 599 (1995) ( citations omitted ).
[14] See Jonathan D. Polkes, White Collar Crime , New York Law Journal, July 7, 2003.
[15] 18 U.S.C., Sec. 1519 (2004).
[16] 148 CONG. REC. S7418 (daily ed. July 26, 2002) (statement of Sen. Leahy).

[18] United States v. Gaudrin, 515 U.S. 506, 509 (1995). [19] http://www.davidconn.com/false_statements.html.(no longer accessible) Linnea B. McCord, JD, MBA,, Associate Professor of Business Law at the Graziadio School of of the Simultaneous Conflicting Emotions By Don Business and Management, Pepperdine University. Dr. McCord started teaching business law and ethics more than 30 years ago, first as an in-house corporate counsel and later as the General Counsel of a division that was part of a high-tech Fortune 500 multinational corporation, headquartered in New York and Paris.

Her area of expertise is the critical role Rule of Law plays in **How Communication Performance in the Work Place**, the long-term success of economies and countries and why American Rule of Law is unique in the world.
The opinions expressed are solely those of the authors and do not necessarily reflect the views of the Graziadio School of Business and **A Description Conflicting Emotions By Don Giovanni** Management nor Pepperdine University.