Cyclotomic Extension

ThePerfectHacker · #1 $Old$ June 17th, 2008, 11:20 PM

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.

flyingsquirrel · #2 $Old$ June 18th, 2008, 12:15 AM

Hi

Here is your PDF file. Let me know if anything has to be modified.

ThePerfectHacker · #3 $Old$ June 18th, 2008, 08:39 AM

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.

kalagota · #4 $Old$ June 18th, 2008, 08:43 AM

without your respect sir, can i ask why you can't do pdf?
i'm just curious.. Ü hope, you won't mind..

ThePerfectHacker · #5 $Old$ June 18th, 2008, 08:47 AM

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.

kalagota · #6 $Old$ June 18th, 2008, 08:54 AM

hahaha.. ok.

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

* > > > OVERJOYED < < < - LaTex.. LaTex.. LaTex

flyingsquirrel · #7 $Old$ June 18th, 2008, 09:11 AM

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.

Krizalid · #8 $Old$ June 18th, 2008, 09:38 AM

You may want to increase the size of the document or modify layout options.

ThePerfectHacker · #9 $Old$ June 18th, 2008, 11:18 AM

Thank you very much, you made it look like a professional written paper.
I am sorry that France lost yesterday.

flyingsquirrel · #10 $Old$ June 18th, 2008, 12:31 PM

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.

Quote:

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.