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  #1  
Old June 17th, 2008, 11:20 PM
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Default Cyclotomic Extension

We had the following problem for homework: let d\in \mathbb{Q} prove that \mathbb{Q}(\sqrt{d}) lies in a cyclotomic extension.
(Without using Kronecker-Weber Theorem!)

I found a solution to this problem using quadradic Gauss sums, however this is not a course in number theory - it is a course on Galois theory, therefore my solution might be not acceptable. I will post it anyway because I want to print it out and give it to the class so which is why I am hanging it on the forum. I am interested in seeing other solutions.
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Last edited by ThePerfectHacker; June 18th, 2008 at 08:38 AM.
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  #2  
Old June 18th, 2008, 12:15 AM
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Hi

Here is your PDF file. Let me know if anything has to be modified.
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  #3  
Old June 18th, 2008, 08:39 AM
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Thank you very much!

There are two mistakes that I did. The first mistake appears in the proof of Theorem 1. The part in the parenthesis "If this fact is not familar ... "
After Theorem 3, "As a result ... ", that paragraph. It should say g_1 = \pm i\sqrt{p} and not g_1 = i\pm \sqrt{p}.

I edited my post above if it makes it any easier.
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Old June 18th, 2008, 08:43 AM
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without your respect sir, can i ask why you can't do pdf?
i'm just curious.. Ü hope, you won't mind..
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Old June 18th, 2008, 08:47 AM
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Quote:
Originally Posted by kalagota View Post
without your respect sir, can i ask why you can't do pdf?
I can do a PDF, but why should I do it if I can find someone else to do it?
Just joking! I never made a PDF before, I do not have a program which allows me to make a PDF, and I do not have LaTeX installed on my computer.
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Old June 18th, 2008, 08:54 AM
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hahaha.. ok.

if you are interested in installing one, try downloading it here..

* > > > OVERJOYED < < < - LaTex.. LaTex.. LaTex
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Old June 18th, 2008, 09:11 AM
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Quote:
Originally Posted by ThePerfectHacker View Post
Thank you very much!

There are two mistakes that I did. The first mistake appears in the proof of Theorem 1. The part in the parenthesis "If this fact is not familar ... "
After Theorem 3, "As a result ... ", that paragraph. It should say g_1 = \pm i\sqrt{p} and not g_1 = i\pm \sqrt{p}.

I edited my post above if it makes it any easier.
I've corrected these two mistakes, see attachment.
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File Type: pdf tph_2.pdf (69.0 KB, 36 views)
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Old June 18th, 2008, 09:38 AM
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You may want to increase the size of the document or modify layout options.
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Old June 18th, 2008, 11:18 AM
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Thank you very much, you made it look like a professional written paper.
I am sorry that France lost yesterday.
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Old June 18th, 2008, 12:31 PM
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Quote:
Originally Posted by ThePerfectHacker View Post
Thank you very much, you made it look like a professional written paper.
Well that's why it's interesting to use LaTeX : once one knows the commands used to write equations (like \frac{}{} ...) it's really easy to make a document like this one.
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I am sorry that France lost yesterday.
It's been a while since I've watched a football match on TV so I won't comment on this.
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