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Ways to Cite a Website in APA - wikiHow

Websites in apa format

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APA Citation Guide: How to Cite a Website in APA - BibMe

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APA Citation Guide: How to Cite a Website in APA - BibMe

Bay Area Career Coach + Life Strategist. Resume #038; Cover Letter Review #038; Refinement. Good idea about goals, interests, demonstrated skills Increasing clarity on personal strengths and weaknesses Focused investment of time, resources, and positive energy! View our special, high-value. SPECIAL VALUE Resume Writing PACKAGES. Websites! (Review, Refinement, Strategic Advice) Total of 2 hours of phone consultation in 2 sessions. Essay About In Ibsen's A Doll's House! Total of 4 hours of phone consultation in 3 sessions.

Total of 6 hours of phone and face-to-face consultation in 4 sessions. Websites In Apa Format! Refine job search strategy, target positions, identify strengths transferable skills from other jobs / projects. Clarify career goals, ideal job positions, interview advice talking points. * Discounts on resume refinement and advisory services: Special discount for students and congress of vienna definition new college grads: 15% For those within 5 years of websites in apa college graduation (undergrad. degree): 10% Special offer for those in between jobs: Pay 75% upfront, 25% after you get a job! Get started on congress, boosting your job search and career today. Contact us with a brief description of your goals, target job/s, current situation, and desired package . I will get back to you soon to schedule an websites in apa, initial phone discussion (free) to go over the process, clarify any issues, and to inferior good graph, ensure we are a good fit. How to in apa, get the best resume critique #038; advice. Sequence Of Moral Development By Lawrence Kohlberg?! The sections below elaborate on the bullet points above highlighting the critical attributes that make a big difference when choosing a resume service. It explains how top resume coaches leverage their knowledge and business experience to develop a better understanding of your goals strengths. Building the necessary foundation that leads to the creation of websites in apa format high quality, effective resumes and more fruitful job search efforts! Vital pre-requisite for reviewing writing a winning resume. Writing / updating a professional resume is what is the sequence kohlberg? not just about language / writing skills.

What makes a difference is websites format good guidance from Idealism and Its, people with considerable industry as well as recruiting experience. Industry professionals with well honed observation skills and good judgment. To create impressive resumes and cover letters that get you the websites in apa format job / career you are aiming for, the best way to start is getting objective, professional feedback. Valuable critique that includes sharp insights and good, actionable ideas and powers tips based on websites in apa format, solid corporate experience . Helping you develop powerful,well-written resumes that effectively present and position your credentials for your dream job. Professional leadership and management experience also brings with it the ability to good graph, provide both positive feedback (on things you are doing well), as well as corrective feedback (on things you can improve) Ė given in a constructive, encouraging, inspiring way. Value of real-world business experience in effective resume development. Resume writing services in the Bay Area are an area where quantity may exceed quality. When you end up with an OK (or worse) service, you do not simply waste a few hundred dollars; the format real cost / loss can be substantial losing out on great opportunities for professional and witch powers financial advancement. Good decision making and in apa judgment (that your employers look for) starts with good personal choices and meaningful investments in witch powers your self-development! For high quality resumes and cover letters that help you get your dream job, the best option is a career coach who understands business priorities and management issues based on their significant work experience.

And who can recognize the relative importance as well as the underlying needs behind the long list of skill and experience requirements on your target job posting. Websites! Professional CV creation and refinement delivers great value for powers, money (ROI) when you are getting resume improvement recommendations that go beyond language / grammar issues and in apa format visual presentation. An outstanding resume influences a hiring manager to continue reading, and then add it to their shortlist. That happens when your resume highlights all your varied strengths and on leadership demonstrated skills and the FIT with the job you are seeking. Top resume writers apply marketing principles for websites, best results. Sequence Of Stages Development By Lawrence Kohlberg?! It is good to websites in apa format, view this as a marketing exercise. You are the product and the goal is to enhance your value and appeal in congress the job-search market where the company / hiring manager performs the buyer role. A resume coach with a thorough professional understanding of: strategic marketing principles (not just promotion) buying process (in this case the format hiring / recruitment process) effective communication / pitch-writing skills. can significantly help you shape the marketing campaign that is and abilities your job search!

For tips about improving the product itself (another core marketing task), i.e. refining your skills, goals, action plans identifying and websites in apa building on graph, your strengths addressing any weaknesses and gaps. In the section below, you can view a summary of the resume review / resume development process. In Apa! To get started on Essay Idealism Struggles, strategically improving your job search efforts, please use the in apa format links to the contact page. The detailed descriptions of the witch major steps of the websites format resume review and refinement process are now on powers, a separate page. You can use the main navigation menus or follow the link below: To get started on this vital step towards your new job, please write to me with a brief outline of your background and goals. I will get in touch shortly with additional details.

High quality, effective, resume coaching / job search coaching services based in Silicon Valley. Serving the entire San Francisco Bay Area and beyond. Major emphasis on the cities of South Bay, East Bay, and Peninsula areas. Format! Primary focus on the high-tech / IT industry, spanning business roles (product management, marketing, finance, business development, general management) as well as technical / software development / IT specialist roles. About Idealism! From new college grads, entry-level positions, to senior management roles. Contact us today with any questions and to get started. Based in in apa San Francisco Bay Area, Northern California. Or if you prefer, Silicon Valley. Palo Alto, Redwood City, Mountain View, Sunnyvale, Santa Clara, San Jose, Milpitas, Fremont, Union City, San Leandro, Hayward, Oakland, Berkeley and other Bay Area cities. You can also reach me at: Career Coach and Strategist with considerable real-world experience in leadership, management, communications at top-tier consulting firms, large leading corporations as well as small businesses, start-ups, non-profits.

MBA, DUKE University. BS, Engineering, IIT. Get started on the 3 Pillars of SELF MARKETING. CLEAR GOALS STRATEGY FOR PERSONAL GROWTH VISIBILITY COMMUNICATIONS ACTION PLANS. San Francisco Bay Area, California - serving South Bay, Peninsula, East Bay, Contra Costa Alameda, San Francisco, CA.

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How do you reference a web page that lists no author? - APA Style

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Simple Software for Better Interview Skills. Learn how to websites in apa land the what is the kohlberg? job with Big Interviewís powerful video tutorials and virtual interview practice software. Cover Letter Examples That Will Get You Noticed. A strong resume cover letter can mean the difference between landing a job interview and getting passed over. Read and live by this comprehensive cover letter guide from our resume expert and professional resume writer Kimberly Sarmiento and check out her cover letter examples for inspiration. You Really Do Need a Strong Cover Letter. You never get a second chance to websites in apa make a good first impression in the job search. And in Essay Use of in Ibsen's Play, A Doll's House, most cases, your first impression on a hiring manager begins with your resume and cover letter. If you donít get the in apa cover letter right, you may never get the opportunity to wow them with your new suit, confident eye contact, and compelling interview stories. Even if you network your way into that job interview (and even if you got a great referral from one of your advocates), the Essay about and Its Struggles to Exist hiring manager will look at in apa, your resume and/or cover letter and use them to form or influence an opinion prior to Essay Idealism and Its meeting you . That is why in my 2009 book, ďThe Complete Guide to Writing Effective Resume Cover Letters: Step by Step Instructions,Ē I refer to websites in apa your cover letter as your handshake and your sales pitch all rolled into one.

I can hear the scoffing now and witch, the protesting that there is format no way a cover letter can be that important. Graph? ďResumes are selected by key word scans or passed off from one contact to another. No one really reads cover letters anymore, right?Ē Wrong. The simple truth is that at websites format, some point in time your resume Ė and graph, your cover letter Ė will be reviewed by a real live person (if youíre lucky). That person will be deciding whether or not you are worth their time to websites format interview and your cover letter can help confirm that your resume goes into the ďyesĒ pile rather than the ďfile for about in Ibsen's House, future openingĒ pile (or the real or digital garbage can). Sure, there are times when a recruiter or hiring manager will skip right over the cover letter and focus on the resume.

But other screeners wonít even look at your resume if the cover letter doesnít get their attention. Why take a chance? Write a strong cover letter and youíll know that youíre doing everything possible to get past the gatekeepers and score an interview. If you are wondering how to write a compelling cover letter, read on for Kimberlyís advice and examples. Three Situations When a Great Cover Letter is Even More Critical. 1) When you need to in apa format include information that should NOT go into the resume. A resume is a formal business document with strict rules that must be followed. These rules include not writing in first person or including personal information like your desire to relocate. However, there are times when you need to communicate this type of information in order to Essay Struggles to Exist make the case for your fit for the position: Example: Your cover letter can be used to communicate your intention to make a transition in your career or move to another city/state.

Recruiters receive thousands of unqualified resumes for every position. They will look at your resume and cover letter and immediately trash them if they donít see a fit ó assuming that you are another one of format those annoying applicants who applies for every job posted. This is always a challenge for career changers and individuals looking to relocate and a good cover letter can make a big difference. Example: Your cover letter can also explain away other aspects of your particular career situation that might not be appropriate to include on your resume. For example, if you took some time away from the work force, but have kept your skills and knowledge up-to-date. Additionally, in some job ads, the company will ask for specific information to be included in your cover letter. On Leadership? This technique is used to make screening easier ó if someone canít follow simple application directions, why waste time on an interview?

Pay careful attention to the information they request and be sure to address it. One problematic area is if they ask for salary requirements to be included in your cover letter. Companies make this request to websites format help them rule out individuals with higher salary requirements than they have budgeted for the position, but it can also lock you into a lower pay range than they might offer you otherwise. However, ignoring the request could disqualify you as well. Ergo, I suggest you research the witch powers and abilities average salary for format, the position you are applying to in the state of the opening and include a range slightly above and good, below that number. There are several sites that have compiled census and other data information to give you a decent estimate of salaries by position in specific cities and in apa format, states (Payscale is a great place to start). So if the average salary of your job is $60K for congress of vienna definition, the location where you live (or want to live), list your salary requirements as $55K to $65K.

Again, no salary information should be included in a resume. I typically donít even include information about bonuses or commissions for in apa, sales representatives (just awards like Presidentís Club or Top 5%) . 2) When you want to reference a network connection. There is no right way to include in your resume, ďOur mutual associate John Smith referred me to this role and says he thinks I will make a great fit for the job opening.Ē That is a reference line reserved solely for the opening paragraph of essay a cover letter. There are multiple ways you can mention a network connection or mutual friend in websites in apa, a cover letter, but such a statement has no place in a resume whatsoever. Note: In professional resume writing, it has become passe to include a list of references on your resume or even the Essay about Symbols in Ibsen's line ďreferences available upon request.Ē Such information takes up valuable real estate on your resume (which should be 1-2 pages max) and it is best to focus on your achievements and format, qualifications instead. Besides, the hiring managers know you will give them references when they request them. Rather than waste space on your resume, prepare a reference sheet with the same header as your resume and give it to the interviewer at the end of your meeting. This sheet should include the first and last name of your references, their titles and company names, city and state, phone numbers, and e-mail addresses if possible. You can even be proactive and have letters of recommendation ready to hand the hiring manager at the end of definition your meeting, but donít send them prior to that initial interview. 3) When you want to in apa emphasize why youíre interested in the company. One way to distinguish yourself as a job candidate is to research the essay company you are interviewing with and talk about things you like or ask questions about the work they have coming up.

This demonstrates your interest in their particular organization as opposed to them being just another job ad you responded to in your desperate attempt to websites format find employment. You can use your cover letter to show that youíve done your homework and see a strong fit with the organization. Within the second or closing paragraphs of inferior your cover letter, you can mention being interested in the specific work the company does, recent grants they have been awarded, a product they recently released, etc. Again, this is not appropriate for inclusion on your resume, but adding it to your cover letter can help you stand out in apa, from the stack of applications the hiring manager is sorting through on the day your resume passes by him/her. How to Write a Great Cover Letter. Hopefully I have convinced you of the importance of cover letter writing or at least how the powers letter can prove useful to you in certain circumstances.

But how do you write a cover letter that will open doors for websites, you? And how do you avoid mistakes that can lead to Idealism and Its to Exist rejection? Please review these five simple rules for ensuring your cover letter leaves the hiring manager excited about meeting you. 1. Websites Format? First and foremost, the letter must be grammatically correct and error free! If you are not a particularly good writer, have someone read and edit the document for you. 2. If printed, the letter should be one page max. The letter should also be printed on high-quality paper just like your resume. In some instances, you might elect to cut and paste a cover letter into an e-mail and attach your resume.

If so, you want the cover letter to be easily read with minimal scrolling. So get to the point and on leadership, be succinct. 3. The letter should include examples of your qualifications. You can write a cover letter in paragraph or bulleted formats, but either way, you should include examples of your achievements and credentials. Websites In Apa Format? While you want to be brief, you also want to witch and abilities encourage the reader to review your resume for greater detail. The best way to do that is to call out two or three things that you have done professionally to catch their attention and websites in apa, make them want to know more about you. Make sure you customize the letter to what of stages of moral by lawrence highlight the achievements most relevant for each position. 4. Your letter should address a specific person. Whenever possible, do some research and find out the websites personís name who will be reading your cover letter. This is a minor detail and on leadership, some hiring managers wonít care, but it can distinguish you from your competition all the same. More importantly, donít send an obviously-generic letter that has not been customized for the company/position.

5. Your letter should end with a call to websites action. When you close your letter, be sure to ask for a meeting. It is obvious that you want an interview when you submit a cover letter and resume, but job hunting is usually helped along with a proactive approach. Therefore, at every point in the application process you should seek to move yourself along to the next stage of consideration. The cover letter is the first instance of this, so donít miss an opportunity to encourage a meeting with the hiring manager at the close of your letter. Congress Of Vienna Definition? Also be sure to thank them for in apa format, their time and consideration. Check out Kimberlyís cover letter examples to see and learn from the methods that have worked for her resume writing clients. Cover Letter Example 1: Returning to Work after a Job Gap/Relocation. Dear Principal Townson: With five years of about Idealism and Its Struggles to Exist experience in websites in apa format, teaching high school and a masterís degree in Chemistry, I believe I am an ideal candidate to fill the of stages of moral development by lawrence kohlberg? science teaching position you have open with the retirement of Stacy Jones.

My teaching experience was at format, John Smith High School in Smallville, NJ before my husband and I moved here seven years ago. Essay? While there, I taught all levels of Chemistry and helped host the science fair each year. When we moved here, I was pregnant with my oldest. Now that my youngest has started kindergarten, I am eager to in apa format return to the workforce. Witch Powers? Although I focused on my family these last seven years and have not worked for pay, I kept up with developments in teaching and chemistry by websites in apa, reading literature and what development as identified, attending conferences hosted by the American Association of High School Science Instructors. I have also volunteered my time at the community center, tutoring all level of students in general sciences. I look forward to raising my children in this community and someday teaching them at Rosewood High School. Please review my attached resume.

I will be contacting you next week to schedule an websites in apa, interview. Essay About And Its? Thank you for websites, your consideration. Pamís Take: I love how this cover letter emphasizes the applicantís relevant qualifications in the first line. This puts the emphasis on her ability to do the job and not the fact that sheís returning to Essay about Struggles work after several years as a stay-at-home parent. Websites? Later, she briefly explains her break and essay, how she has kept current.

Her resume will clearly show a gap, so it makes sense to proactively address it. Cover Letter Example 2: Transitioning Careers. Blending a formal background in marketing with proven success in websites, retail sales and what is the kohlberg?, customer service roles, I am looking to transition into public relations and websites in apa format, believe I would make a great fit for Essay Symbols in Ibsen's Play, House, the advertised position of Public Relations Specialist at your company. Having both used and sold your products, I am already well versed in websites, your brand and Essay about Use of Play, House, both present and past yearsí offerings. Websites Format? I have followed with excitement as you launched in European and Asian markets and incorporated an international feel into your product line. Of Stages Development By Lawrence? I would bring both passion and in apa format, expertise to championing your company with the press and public. I am already trained in creating buzz and witch and abilities, awareness through social media channels including Twitter and Facebook. When my sales team decided to participate in websites format, Walk for a Cure three years ago, we used social media to heighten our sponsorship support and raised more than $20K for the event. Confident my transferable skills make me a solid candidate for this opening, I respectfully submit my resume for your review and request a meeting to of vienna discuss the opportunity further. Websites In Apa? I will make myself available at your convenience and look forward to witch and abilities your call to arrange a time.

Thank you for your time and consideration. Pamís Take: This candidate leads off with a strong statement about her career change goal and websites, her fit for the specific position at hand. She uses the graph rest of the letter to discuss her interest in the company and websites in apa, some of her key transferable skills/experience. Cover Letter Example 3: Entry Level. I am writing in response to what of moral by lawrence your listing in the Memphis Gazette for a nurseís aide. Please accept my enclosed resume for consideration. As a CPR-certified lifeguard and a LPN student at Memphis Community College, I have the formal training necessary for this position. Additionally, with two years of experience in in apa format, retail sales, I have excellent customer service skills that can translate well to patient relations. After you have reviewed my resume, I hope to meet with you to and abilities discuss how I can be beneficial to your team. I look forward to hearing from you to schedule an interview at your earliest convenience.

Pamís Take: Nice concise approach for an entry-level candidate. She doesnít have years of nursing experience to point to, so she highlights her training and how her non-nursing work experience has also helped to websites in apa prepare her. Cover Letter Example 4: Professional. As an of vienna, Accounts Payable Receivable Specialist , I offer a proven ability to accurately process invoices, payments, reimbursements, and format, tax reports. I quickly learn and adapt to is the sequence of stages of moral as identified by lawrence kohlberg? software changes and updates and help team members resolve issues and problems they are having with data input and processing. Examples of my accomplishments include: ē Handled biweekly accounts payable processing of checks and ACH payments; reconciled payments made to websites format accounts payable software and addressed any discrepancies that arose. ē Created a spreadsheet that listed bank and routing numbers to expedite processing of about Idealism expense reports.

ē Uploaded and websites, reconciled monthly phone bills for what of stages as identified by lawrence, approximately 200 branches and 4 operational centers; total billing amounts were coded for various departments and in apa, branches as required. Confident I will prove valuable to your company, I respectfully submit my resume for your review. Of Vienna Definition? I would also like to websites request a personal meeting to discuss your upcoming goals and how I can help you achieve them. I will make myself available at Symbols in Ibsen's Play, A Doll's House, your convenience and look forward to your call. Thank you for your consideration. Pamís Take: For an experienced candidate, a bit more detail is expected. This candidate customized the bullet points to specifically communicate his experience with the position responsibilities listed in the job description. As a Human Resources Manager with a strong customer service background, I offer expertise in employee relations, benefits administration, and generalist duties. I have made significant contributions in websites, succession planning and workforce engagement as well as ensuring compliance with employment and labor requirements.

I am also known for what is the sequence of stages as identified by lawrence kohlberg?, my ability to help identify and implement key technology and process improvements. I am well-versed in Six Sigma methods and in apa, have lead projects which produced significant and sustainable savings. Other examples of my work include: ē Creates positive employee engagement for graph, 2,000+ personnel at websites format, Company XYZ via proactive communications, prompt issue resolution, and fair/equitable treatment. ē Led Six Sigma project related to FMLA administration and about Symbols House, online orientation programs for Lean Belt training. ē Proved instrumental in the deployment of an E-recruitment system that serviced a Fortune 200 company; defined policies, procedures, and communication planning for the project. Confident I will make a positive impact on your organization, I respectfully submit my resume for your review. I would also like to request a personal meeting to discuss your goals for this position and my potential contributions. Websites In Apa? I will be available at your convenience and look forward to your call to arrange a time. Thank you for your consideration. Pamís Take: This cover letter highlights the applicantís relevant accomplishments as a leader and witch powers and abilities, manager.

It goes beyond stating familiarity with the required job duties and emphasizes results in key projects. Websites? Remember that you donít want to copy and paste your whole resume into the cover letter. Think about the key selling points that you want to feature prominently. The goal is to essay make them excited to learn more about you. Cover Letter Example 6: Senior-Level Executive. As a Senior-Level Finance Operations Executive , I offer proven success in format, maximizing productivity and improving profit margins. My work spans companies and business units at essay on leadership, various stages of growth, including start-up, established, and turnaround settings. Believing profitability requires strong revenue generation and cost controls, I monitor budgets and sales performance closely to identify areas for improvement. I am known for enhancing overall performance through technology upgrades, advanced employee training, and implementation of best practices.

Examples of my work include: ē Drove successful launch of start-up company by hiring a talent team, defining product development plans, and leading go-to-market strategies to achieve $35M+ revenue and 50% margins within two years. ē Managed daily operations of a $150M subsidiary that provided a complete suite of manufacturing solutions in websites in apa format, the US and what is the sequence as identified by lawrence, Canada; improved profit margins on overall product line 10%. ē Proved vital to reorganization leadership that cumulated in a 10% productivity improvement in the sales and websites in apa format, service organization and essay, a 20% improvement in support organizations. As CPA and websites format, MBA, I am confident I will prove valuable to your company and Idealism and Its, respectfully submit my resume for your review. I would also like to request a personal meeting to discuss your upcoming goals and how I can help you achieve them. I will make myself available at websites in apa format, your convenience and look forward to congress definition your call. Websites Format? Thank you for your consideration. Pamís Take: This cover letter nicely distills years of experience into a concise overview that really ďsellsĒ achievements most relevant to what is the of stages development by lawrence kohlberg? the specific advertised role. Each bullet presents a compelling high-level overview of a specific position, complete with impressive data points. Itís hard to be this concise when talking about in apa a long career! However, a concise letter is always more effective ó make the most exciting information jump out Essay about in Ibsen's Play, A Doll's, of the letter and grab the websites in apa format recruiterís attention.

Many thanks to Kimberly for her expert advice and cover letter examples! Have other thoughts on what is the as identified kohlberg? what would make a great cover letter? Leave a comment below. Humor: Mac and Charlie from in apa ďItís Always SunnyĒ teach us an important lesson about what NOT to include on your resume. Enjoy!

Pamela Skillings is about co-founder of Big Interview. As an in apa, interview coach, she has helped her clients land dream jobs at companies including Google, Microsoft, Goldman Sachs, and JP Morgan Chase. She also has more than 15 years of what is the sequence of moral experience training and advising managers at format, organizations from American Express to the City of powers New York. She is an adjunct professor at New York University and an instructor at the American Management Association. 14 Comment to format Cover Letter Examples That Will Get You Noticed. Good article, Pam. I agree totally with points 2 and 3. I recently was coaching someone on how to use step 2. If you were submitting just a resume, like I know a lot of applicants do, it would seem very awkward to include that reference somewhere in the text of the Essay about Use of Play, resume.

It is websites good to include that in the first couple sentences to make the personís name stand out. Who you know goes a long way in the career field! Thanks for the comment! Yes, itís amazing the difference that a referral can make. very helpful, I have been applying but never landed even a single interview. I have applied so many jobs online but i have never got even one call for Essay about Idealism and Its to Exist, interview. Thank you very much for this information, I really liked the points 1 and 3, I consider this information very clearly exemplifies the elements that must be considered to make a cover letter. This is an excellent phrase: You never get a second chance to make a good first impression in the job search.

Thanks. Your suggestions are concise and helpful for the many of us out here sending in application after application. What is your opinion on how to show that you are attentive to detail (one of websites in apa job responsibilities for a current job opening) . I was thinking that I could point out some typos on is the development as identified by lawrence kohlberg? the companies website. Do you think this would be taken as a criticism rather than a helpful suggestion? Great website! Thank you! These are all very helpful examples! I do have one question: It seems unconventional ó and a bit overly assertive Ė to jump right in on the the first sentence of the format letter without introducing yourself by name. The standard: My name is ó- and I am applying for the ó- position.

Is it really ok to do this these days? Excellent written and well explained. As a recruiter I honestly donít spend much time reading cover letters. Hiring managers have to what is the of moral as identified sort through hundreds of resumes and realistically donít have time to give most resumes and cover letters more than 30 seconds to scan for the information they are looking for. The cover letter should be short, sweet and websites in apa format, highlight the key points you want us to know. Regarding the comments above about applying for jobs but not getting interviews.

The resume is the most important part of the application and must be well written with clean formatting. Recruiters and HR Managers want to get to of moral by lawrence kohlberg? the point of where have you been working, for how long and what have you accomplished? Also, you may want to tailor your resume and in apa format, cover letter to each job you are applying to of stages of moral as identified so that it shows to be a good match for the job. Format? Furthermore, you should do more than just apply for a job online you need to network with recruiters, attend industry trade shows, join group discussions on Linkedin and more to get you name out there. Love your site. In Example 5 above, you have a misused word. It says, ďÖand have lead projects which producedĒ when it should use ďledĒ, not ďleadĒ as lead is current tense and led is sequence by lawrence past tense. your comment is so helpfulÖ thank you.

Why do you put a comma after dear but also a colon? What is the correct punctuation for in apa format, a formal cover letter? Do you think itís appropriate to arrange a time on the cover letter to call the employer so that you can schedule an definition, interview with them? Those letters are really engaging and a lot creative. Websites In Apa? A project engineer or project manager will list projects successfully handled as achievements. Pamela Skillings is co-founder of Big Interview. As one of the country's top interview coaches, she has helped her clients land dream jobs at of vienna, companies including Google, Microsoft, Goldman Sachs, and JP Morgan Chase.

She also has more than 15 years of format experience training and advising managers at organizations from American Express to the City of New York. She is an good graph, adjunct professor at New York University and an instructor at the American Management Association. Continue reading. Copyright © 2017 Big Interview Ė Job Interview Training - All Rights Reserved. Skillful Communications, LLC | 244 Fifth Avenue, 2nd Floor | New York, NY 10001 XML SItemap.

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Websites - APA Referencing Style Guide - Library Guides at AUT

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Websites - APA Referencing Style Guide - Library Guides at AUT

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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. About Use Of Symbols In Ibsen's A Doll's House? 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. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . . . . . . . . . In Apa? . . . . . . . Essay On Leadership? . . . . . . . . . . . . . . . . . . . . . 5 Prerequisites . . Format? . . . . . . . . . . . . . . . . Witch Powers? . . . . . Websites In Apa Format? . . . . . . . . . . . . . . 5 Acknowledgements . . . . . . . . . . . . . . . In Ibsen's Play, House? . . . . . . . . In Apa? . . . . . . . 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Inferior Good? . . . 97 Units in cyclotomic fields . . . . . . . . Websites Format? . . . . . . . . . . . . . . . . . . . Congress Of Vienna? . . . 97 The first case of Fermatís last theorem for regular primes . . . . . . . . . . . . . 98 Exercises . . . . . . . . . . . . . . . . . . . . . . . In Apa Format? . . . . . . . Inferior? . Format? . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . . . Essay On Leadership? . . . . Websites? . . . . . . On Leadership? . . . . . . . . . . . . . . . . . . . . . Websites In Apa? . . . . 101 Nonarchimedean valuations . . . . . . . . . . Congress Of Vienna Definition? . . . . . . . . . . . . . . . . . . . 102 Equivalent valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . . . . . . . . . . . . . . . Websites? . 105 Complete list of valuations for the rational numbers . . . . . Congress Definition? . . . . . . . . . . . 105 The primes of a number field . . . . . Websites In Apa? . . . . . . . . . . . . . . . . . . . . . . . 107 The weak approximation theorem . . . . . . . . . . . . . . . . . . . . . Essay On Leadership? . . . . 109 Completions . . . . . . . . . . . . . . . . . . . . . . Format? . . . . Essay Idealism Struggles To Exist? . . . Websites Format? . . . . . . . . 110 Completions in the nonarchimedean case . . . . . . . . . . . . . . . . . . . . . . 111 Newtonís lemma . . . . . . . . . . . . . . . . . . . On Leadership? . . . . Websites In Apa Format? . . . . Essay About Use Of Symbols In Ibsen's A Doll's? . . . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . . . . . . . . . . Websites Format? . . . . 118.

Newtonís polygon . . . . . . . . . . . . . . . Of Vienna? . . . . . . . . . . . . . . . . . . . 120 Locally compact fields . . . . In Apa Format? . . . . . . . . . . . . . . About In Ibsen's A Doll's? . . . . . . . . . . . . . 122 Unramified extensions of websites format, a local field . . . . . . . . . . . . . . . . . . . . . . . 123 Totally ramified extensions of K . . . . . . . . . . . . . . . . . . What Is The Sequence Of Stages Of Moral Kohlberg?? . . . . . . . . 125 Ramification groups . . . . . . . . . . . . In Apa Format? . . . Essay About Use Of Symbols In Ibsen's? . . . . . . . . . . . . Format? . . . . . . 126 Krasnerís lemma and applications . . . . . . . . . . . . . . . . Essay About And Its To Exist? . . . . . . . . . 127 Exercises . . Websites In Apa? . . . . . . . . . . . . . . . . And Abilities? . . . . Format? . . . . Witch And Abilities? . . . . . Websites In Apa? . . . . . . . 129. 8 Global Fields 131 Extending valuations . Inferior Graph? . . . In Apa? . . . . . . . . . . . . . . . . . . . . . . . . . . . Essay About Use Of A Doll's House? . 131 The product formula . . In Apa? . . . . . . . . . . . . . . . Powers And Abilities? . . . . . . . . Websites In Apa? . . . . . . . 133 Decomposition groups . . . . . . . . . . . . . . . . On Leadership? . . . . . . . . . . . . . . . 135 The Frobenius element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Examples . . . . . . . Format? . . . . . . . . . . . . . . . . . . . Congress Of Vienna? . . . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . . . . . In Apa? . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . . . . . . . . . . . . . . . . . Essay In Ibsen's Play, House? . . . . 141 Applications of the Chebotarev density theorem . . Websites Format? . . . . . . . . . . . . . . . . 146 Exercises . . . About And Its Struggles To Exist? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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. Websites In Apa Format? 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 finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i2I , is and Its to Exist, a function i 7! ai WI ! A. X Y X is in apa format, a subset of congress, 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 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. 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 (1601Ė1665). In Apa? Stated his last ďtheoremĒ, and proved it for mD 4. He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the of vienna definition, 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 (1707Ė1783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. LAGRANGE (1736Ė1813). 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 format, algorithm for solving (1) always leads to a solution, LEGENDRE (1752Ė1833). He introduced the ďLegendre symbolĒ 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 (1777Ė1855). 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 (1805Ė1859). 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. He proved the following ďunit theoremĒ: 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 good graph, a finitely generated group of rank rC s?1. KUMMER (1810Ė1893). He made a deep study of the arithmetic of in apa format, cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of ďideal numbersĒ. He proved that Fermatís last theorem holds for regular primes.

HERMITE (1822Ė1901). 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 (1823Ė1852). He published the first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (1823Ė1891). He developed an alternative to Dedekindís ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum).

RIEMANN (1826Ė1866). Studied the Riemann zeta function, and made the Riemann hy- pothesis. On Leadership? 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (1831Ė1916). He laid the modern foundations of algebraic number theory by finding the correct definition of the websites format, 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 (1842Ė1913). Made important progress in class field theory and the Kronecker Jugendtraum. Use Of Symbols Play, A Doll's? HENSEL (1861Ė1941).

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 (1862Ė1943). He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of websites in apa, his famous problems were on number theory, and have also been influential. TAKAGI (1875Ė1960). He proved the fundamental theorems of witch powers and abilities, abelian class field theory, as conjectured by websites in apa Weber and Hilbert. NOETHER (1882Ė1935). 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 (1887Ė1947).

Introduced HeckeL-series generalizing both DirichletísL-series and Dedekindís zeta functions. ARTIN (1898Ė1962). He found the ďArtin reciprocity lawĒ, which is the main theorem of class field theory (improvement of Takagiís results). Introduced the Artin L-series. HASSE (1898Ė1979).

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 (1901Ė1977). Defined the congress of vienna, Brauer group, and contributed to the classification of websites format, central simple algebras over number fields. WEIL (1906Ė1998). Defined the Weil group, which enabled him to give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (1909Ė84). The main statements of class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. Essay On Leadership? IWASAWA (1917Ė1998). He introduced an in apa format, important new approach into algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925Ė ). 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 (1936Ė ). Essay About Symbols In Ibsen's House? 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 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 websites format, 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 essay, 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 websites in apa, unique up to 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 Essay about Use of Play, A Doll's makes sense with respect to a subring, and so we must define the ďring of integersĒ OK in 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. Finally, since factorization is only considered up to units, in order to fully understand the arithmetic of K, we need to understand the structure of the group of units UK in OK . 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 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 difficult to apply. We shall show (2.11) that ? is an algebraic integer if and websites in apa format only if its minimum polynomial over Q has coefficients in and abilities, Z. Consider for 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. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and so ? is an algebraic integer if and only if. 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 only if a and b are either both integers or both half-integers, i.e., For example, the 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 canít be written as a product of two nonunits. 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 websites format, essentially one way. Is the what is the by lawrence, 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 in apa format 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 it is easy to see that, for ? 2OK , Nm.?/D 1 Ē ? N? D 1 Ē ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases donít occur, and in is the sequence of stages of moral development by lawrence kohlberg?, the fourth case ? is a unit. A similar argument shows that 2;3, and 1?. 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. Format? OK need not be prime. Essay? 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. 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 format, enlarge the set of ďprime numbersĒ 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 essay, above factorization; here the pi are ďideal prime factorsĒ.

How do we define ďideal factorsĒ? Clearly, an 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 ďprime ideal factorĒ. Since all we know about an ideal factor is the set of websites, elements it divides, we may as well identify it with this set. Thus an ideal factor a is is the by lawrence, 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 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 in apa format, 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?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Since every gen- erator is 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. How far is this from what we want, namely, unique factorization of elements? In other. words, how many ďidealĒ elements have we had to add to our ďrealĒ elements to essay on leadership, 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 ďfractionalĒ 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 Ē OK is a principal ideal domain Ē 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 Fermatís last ďtheoremĒ, 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 ďinfinite descentĒ, 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 in apa, 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 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 about, 1847 in format, which he claimed to witch powers, prove Fermatís last theorem using the following ideas. Let p 2 be a prime, and websites 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 hence a contradiction. Liouville immediately questioned a step in Lame?ís 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? couldnít justify his step (Z?? is not always a principal ideal domain), and Fermatís last theorem was not proved for almost 150 years. However, shortly after Lame?ís embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermatís last theorem for 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 polynomial f .X/ 2Q?X? is a Galois extension of Q. In a basic Galois theory course, we learn how to compute the powers, Galois group only when the websites format, 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. Cohenís book is the more useful as a supplement to this course, but wasnít 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 ďgoodĒ 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 Essay Idealism and Its Struggles to Exist, a history of algebraic number theory, con- centrating on the efforts to prove Fermatís 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 ďideal numbersĒ, 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 ring of integers in Q? p d? is in apa format, as described. 0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is inferior good graph, a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from Commutative.

Many results that were first proved for rings of integers in number fields are true for more general commutative rings, and it is more natural to prove them in 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 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 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 in apa format when B D A??1; . ;?m? we say that the ?i generate B as an A-algebra. For elements a1;a2; : : : of powers and abilities, A, we let .a1;a2; : : :/ denote the smallest ideal containing the ai . It consists of finite sums. Websites Format? P ciai , ci 2 A, and it is inferior graph, called the ideal generated by. a1;a2; : : :. In Apa? 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 ó in fact, it is the smallest ideal containing both a and b. 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 powers and abilities, 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 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 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 ó if A is Noetherian (see below) this is obvious; otherwise the proof requires Zornís lemma.

In particular, every nonunit in A is contained in a maximal ideal. There are the implications: A is 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 websites in apa format, AB is 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 .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 inferior good, 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 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 . (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 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 websites in apa 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 a maximal element after only finitely many steps. (c) (a): Let a be an ideal in A, and let S be the set of inferior good, 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 properly containing a0. Websites? This contradicts the congress of vienna definition, definition of a0. 2. A famous theorem of Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the websites format, 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 and abilities, 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 integral domain A, .a/ .b/ Ē bja, with equality if and only if b D aunit: The first assertion is websites in apa format, 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 witch and abilities, statement of the in apa format, 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 good graph, irreducible elements, it is not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals canít be equal for otherwise c would be a unit. From the maximality of .a/, we deduce that b can be written as a product of irreducible elements, and similarly for in apa format c. Thus a is a product of irreducible elements, and essay 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 ó an algebraic integer ? can not be an irreducible element of O because. Websites Format? p ? will also be. an algebraic integer and ? D p ? p ?. Thus O is 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 an A-module M are equivalent: (a) M is Noetherian; (b) every ascending chain of and abilities, 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 M . 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 websites format, 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 . Graph? Then x?y 2M 00N M 0, and so x 2M 0. Websites? 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. Idealism To Exist? 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 websites 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). LEMMA 1.9 (NAKAYAMAíS 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. 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 essay on leadership, show that a.M=N/DM=N for then (a) shows that M=N D 0. Con- sider mCN , m 2M . 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 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 k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. A subset S of A is in apa format, said to be multiplicative if 0 Ö S , 1 2 S , and S is closed under multiplication. If S is Essay, 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. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. In Apa? 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 Essay about Use of in Ibsen's House, 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. Format? 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. And Abilities? PROPOSITION 1.11 Consider an 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 . 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 websites in apa, 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 sequence development as identified by lawrence kohlberg? so at least one of a=s or s is in p (because it is prime); but s Ö 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 format, S?1A; the inverse map is p 7! pA. PROOF. It is easy to about Idealism and Its Struggles, 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 in apa format, a prime ideal in A disjoint from S; and graph (1.11) shows that the websites format, two maps are inverse.

2. EXAMPLE 1.13 (a) If p is of vienna definition, a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). Websites Format? (b) We list the prime ideals in congress of vienna, 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. In Apa Format? 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. 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 terms of powers, ideals. Integersm and 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 in apa format, 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 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; . Witch? ;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. Websites In Apa? We can now apply the theorem in the case nD 2 to of vienna, obtain an element y1 of 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 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 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; . ;an be ideals in websites in apa, 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 each of the following two conditions: (a) The mapM N !M ?AN is A-bilinear, and good graph any other A-bilinear mapM N ! P is of the in apa format, 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. What Is The Sequence Of Moral Kohlberg?? 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. By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and websites in apa format som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the Essay Use of Symbols in Ibsen's Play, A Doll's House, image of websites format, ??? contains a set of generators for M 0?AN 0 and 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. Essay Symbols Play,? with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an 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 ís with coefficients in A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of 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. In Apa Format? 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 good, isomorphism. M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS.

If A! B is an A-algebra and websites in apa format 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 witch powers, B-module obtained from M by extension of scalars. In Apa Format? 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 a free A-module with basis e1; : : : ; em, then B?AM is essay on leadership, 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 websites, 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 congress of vienna, 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 isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Because K is websites in apa, 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. Finally, ??X?=.fi .X// is a finite separable field extension of ? of degree degfi . Thus we have proved 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 powers, assume in (1.18) that K is separable over k. Websites Format? If not, there will be an ? 2K such that ?p 2 k but ? Ö k, and the ring K?kK will contain an about Play,, 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 canít be a product of websites in apa format, fields. NOTES Ideals were introduced and studied by Dedekind for rings of algebraic integers, and later by others in polynomial rings. About Play, A Doll's? It was not until the 1920s that the theory was placed in its most natural setting, that of arbitrary commutative rings (by Emil Artin and Emmy Noether). Embed this document on your website. If you don't receive any email, please check your Junk Mail box.

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General Histories of Astronomy and format Cosmology. Since the dawn of time there have been astronomical (more generally called heavenly) phenomena which not only ‘forced’ our human ancestors to look towards the heavens but also affected their lives in great many ways. To mention a few of these phenomena, one should point to the never- ending succession of inferior graph night by day and vice versa, the four seasons spanning across a whole year, the bright stellar constellations covering the websites in apa, whole of the night sky. To be a bit more specific, the four seasons would affect the humans nutrition while the stellar constellations would facilitate navigation around the then-known world (hence increase trading and the profits made from it). Many of the very first civilisations (Egyptians, Assyrians, etc.) glorified heavenly objects such as the witch powers, sun or the moon to a point where they considered them as the format, upper gods in their deities’ pantheon. One of the first applications of astronomy appearing in the history of human civilisation (circa 4000-1000 BC) is the adaptation of the moon calendar to the solar year which was a bit larger than the corresponding lunar periods. This adaptation was realised for inferior graph, agricultural reasons. Websites In Apa. Another characteristic example of a civilisation adapting its calendar to its agricultural needs is the Jews (their calendar became solar based instead of the lunar based calendar they originally used). Another ancient civilisation whose existence is closely linked to astronomy and its effects were the Babylonians . Sequence Of Stages Development As Identified Kohlberg?. Their measurements regarding appearance and websites in apa format disappearance of heavenly bodies and the extremely accurate position monitoring of other bright planets of our solar system (e.g Venus) righteously gives to inferior the Babylonians a prominent place amongst the pioneers of the astronomical science.

The Assyrians on the other hand were more cruel and crude compared to the Babylonians even if these two share more or less the same time slot in human history. The Assyrians treated the sun and the moon as deities like many of the Mesopotamian people at the time, but also demonstrated a vibrant interest in in apa, astrology and not so much in astronomy . One of the primary reasons that they used a calendar was not for astronomical observations or other reasons, but for astrological predictions; Assyrians deeply believed that the essay, effect of the format, stars and Essay heavenly bodies on in apa format the behaviour and thought of the man was quite significant. What Sequence Of Stages As Identified By Lawrence. What is even more interesting is that Assyrians in their effort to discover and websites interpret omens regarding kingdoms, kings, empires etc., observed more closely the moon and the stars rather than the sun. This ‘preference’ was due to the fact that the sun does not present any variations in his colour or trajectory throughout the Essay about Play, A Doll's House, time that he is observable from the earth. Websites In Apa. The planets and the moon in the dark sky on witch powers the other hand appear to have multiple trajectories (relative to the earth of course) depending on the time of year, etc. The Assyrians’ kingdom however was dismantled by a combined effort of the Babylonians and their allies in the late 7th century BC. This conquest marks a new era for the science of astronomy. No more omens are exclusively sought after in the night skies, but scientific observations and mapping efforts of the websites in apa, night sky and its objects is beginning to be realised. The town of Babylon itself which is the centre of this cultural and Essay Struggles scientific revolution will pass from the hands of many conquerorsstability however is ensured throughout those transitions and this city will become the epicentre of this astronomical evolution up until the time of the deat of Alexander the Great roughly three centuries afterwards. However, it should not be forgotten that the main driving force behind the websites in apa, evolution of astronomy in inferior, Babylon were the priests who, locked inside their temple they made their observations but this situation had one inherent disadvantage: their knowledge and scientific methods are not documented anywhere; instead, the knowledge passed on from templar to templar most likely.

In the years after the decay of Babylon, the Chaldeans appear and take over further advancing the astronomical science. Greek mathematicians at the time also use for the first time elliptical coordinates systems to fully describe the motion of a planet inside space and websites in apa not only on what of stages kohlberg? a two-dimensional area (as the Babylonian priests did up to that time). Websites In Apa Format. The Chaldeans created the now-famous Chaldean tables which are essentially a whole collection of various data on motion of the planets inside a 3-D elliptical world which is cut in two parts: an upper and a lower part which is smaller. The people with the greatest influence on the evolution of astronomy during the antiquity were without doubt the Greeks. In classical plays such as the Iliad and essay the Odyssey, Homer clearly mentions a series of constellations and stars known at the time such as the Great Bear, Orion, etc. Websites. Some of the of vienna definition, early settlers of Greece were also forced to migrate to the islands of the Aegean or to Asia Minor due to the harsh morphology of mainland Greece; a side effect of this migration were the commercial bonds developed between those various towns.

Commercial bonds means trading, and websites in apa trading also comprises accurate navigation form town to town; these were typical needs which lead to the evolution of Astronomy in ancient Greece. The Greeks on the other hand still worshipped heavenly bodies as deities like the rest of the people of their time. Early Greek philosophers have tried to give some reasoning to phenomena such as eclipses, day and night, etc. Their reasoning however hardly had any practical value. The first to express an opinion of value was Thales who foretold within a year an on leadership, actual solar eclipse.

It is worth noticing at this point that a few Greek philosophers were the in apa, very first to express the opinion that the earth is of spherical shape and the moon rotates around it: they were Timaeus, Aristoteles and Plato. The latter’s theory was based on two sentences firstly expressed by the two former. The fact remains that by graph the 4th century BC, astronomy had evolved in Greece up to a point where for example the elliptical trajectories of individual planets were easy to be observed throughout the in apa, year; this evolution however was nothing compared to the corresponding revolution of other arts and philosophy of the time. In the second century A.D Ptolemy extended the idea of the ancient Greeks for the universe and he proposed a model where the earth was stationary at the center surrounded by eight spheres which carries the sun, the inferior, moon, the five know at that time planets and the stars. The planets moved on their respective spheres and the stars which were at the last and eighth sphere, would always stay at the same positions relative to each other and in apa rotate across the sky; beyond those stars it wasn’t clear what existed. Ptolemy’s model was accurate to predict the Essay about Play, House, position of the planets, while it fails to describe accurately the orbit of the moon. This model has been generally accepted, and since there was a “free’ space beyond the last sphere for heaven and hell this model has also been accepted by the Christian church. In 1514 Nicholas Copernicus proposed a model for the universe but in order to avoid to be characterized as heretic by the church he published it anonymously.

According to his model, the sun was stationary at the center and the earth and the planets where moving around it in circular motion. The discovery of the websites in apa, telescope a century later makes possible the acceptance of Copernicus model. The Italian astronomer Galileo Galilee observed that Juniper’s moon was orbiting around Juniper and witch not as according to Ptolemy’s model around the earth, while the German astronomer Johannes Kepler observed that planets move around sun in elliptical and not circular orbits. Those observations confirm Copernicus model for the universe, but the websites in apa, explanation for the elliptical and not circular motion of the planets came later in 1687. At this time a great publication came to life from Sir Isaac Newton who in his published work explained the motion of the objects with time and he put mathematical expression for those, as well as defining the law of universal gravitation.

According to him, each body in the universe will be attracted by any other body by a force known as gravity; this force is proportional to the mass of the bodies and inversely proportional to the square of the distance between them. Newton’s law of congress gravity gave proof that the websites in apa format, orbits of the planets around the sun will be elliptical as so the orbit of the moon around the earth. Essay In Ibsen's Play, A Doll's. The law of gravity gave a reasonable explanation on the motion of the planets but created a paradox at the same time; since gravity is an attractive force therefore will it cause the planets to collide at some point with each other? Newton argued on this by defining the universe as static and infinite; at websites format that point, no one thought about the possibility that the essay on leadership, universe will expand or contract, but they where trying to modify the law of gravity in order to complete the “picture”. The way we think about the universe changed in 1929 after a great discovery made by Edwin Hubble.

Hubble observed distant stars and galaxies and he discovered that galaxies are moving away from us, or in general the universe is expanding. For an websites, expanding universe there was a time where the universe was still infinitesimally dense and infinitely small, this time has been named as the Big Bang. At that point in time the general laws of physics broke down, and if there was time of the universe before the big bang, we are not able to know it since there is nothing left from them. The idea of a static universe collapsed! The discovery of the electromagnetic theory by James Clerk Maxwell in 1865 brought the idea that light is a wave disturbance of the electromagnetic spectrum , which travels at fixed speeds. Albert Einstein introduced that in inferior good graph, the theory of format relativity in 1905, according to his theory all the laws of physics are the same independently from the speed of the observer. Essay Idealism And Its Struggles. The law of relativity put an end to the idea of absolute time, since it confirmed both Newton’s and Maxwell’s theories. Websites. In 1915 Einstein extended his theory and produced the general theory of about Use of Symbols in Ibsen's relativity; he found out that the universe is a four dimensional space where time is the fourth dimension, thus the space-time universe is not flat but it is curved by websites in apa the disturbance which is caused by the mass of any object. The force of gravity is the what is the by lawrence kohlberg?, product of the format, disturbance of the space-time universe caused by any mass.

For example the mass of the of moral development as identified kohlberg?, sun curves the space-time domain while the earth is moving in websites format, a straight path in Essay about and Its Struggles to Exist, the space time domain it appear from earth (a three dimensional space) that is orbiting in in apa format, an elliptical path. According to the law of relativity, the path of the light is affected by witch powers and abilities the wrapped space time universe; it bends its path so it doesn’t travel in straight line. When the light from websites, a distant star passes near the sun the light is definition, bended because of the in apa, distortion of space time universe by the mass, so for an observer on earth the star will appear in a different position in the sky, the is the of stages of moral as identified kohlberg?, prediction of general relativity confirmed by the observations made in 1919 measuring the position of a star during an eclipse. Websites In Apa. Before the theory of general relativity space and graph time was thought as static, events happening there wouldn’t have an affect on them, after 1915 things changed the universe defined as a dynamic quantity of space and time where each action taken place affects it. A. Websites In Apa Format. Pannekoek, “A HISTORY OF ASTRONOMY ”, Ruskin House, George Allen and Unwin Ltd. J.D. North, “ the Fontana history of Astronomy and Csmology”, Fontana Press I.D. Novikov, “Evolution of the Universe”, Cambridge University Press James Cornell and Alan P. Lightman, “ Revealing the Universe, Pediction and Proof in Astronomy ”, The MIT Press S.W.Hawking, “A Brief History of congress of vienna Time”, Bantam Press 1988 J.J.C. Smart, “Problems of websites Space and Time”, Macmillan Pub. Co, 1964 S.F. Mason, “A History of Sciences”, Colier, 1962 M. Kaku J. Thompson, “Beyond Einstein”, Oxford University Press, 1999.

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