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Math Help - Cyclotomic Extension

  1. #1
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    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.
    Last edited by ThePerfectHacker; June 18th 2008 at 06:38 AM.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    Here is your PDF file. Let me know if anything has to be modified.
    Attached Files Attached Files
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  3. #3
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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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  4. #4
    MHF Contributor kalagota's Avatar
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    without your respect sir, can i ask why you can't do pdf?
    i'm just curious.. hope, you won't mind..
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  5. #5
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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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  6. #6
    MHF Contributor kalagota's Avatar
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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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  7. #7
    Super Member flyingsquirrel's Avatar
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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.
    Attached Files Attached Files
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  8. #8
    Math Engineering Student
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    You may want to increase the size of the document or modify layout options.
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  9. #9
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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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  10. #10
    Super Member flyingsquirrel's Avatar
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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.
    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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