# Thread: Cyclotomic Extension

1. ## Cyclotomic Extension

We had the following problem for homework: let $\displaystyle d\in \mathbb{Q}$ prove that $\displaystyle \mathbb{Q}(\sqrt{d})$ 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.

2. Hi

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

3. 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 $\displaystyle g_1 = \pm i\sqrt{p}$ and not $\displaystyle g_1 = i\pm \sqrt{p}$.

I edited my post above if it makes it any easier.

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

5. Originally Posted by kalagota
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.

6. hahaha.. ok.

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

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

7. Originally Posted by ThePerfectHacker
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 $\displaystyle g_1 = \pm i\sqrt{p}$ and not $\displaystyle g_1 = i\pm \sqrt{p}$.

I edited my post above if it makes it any easier.
I've corrected these two mistakes, see attachment.

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

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

10. Originally Posted by ThePerfectHacker
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.
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.