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Math Help - Cyclotomic Fields....

  1. #1
    AAM
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    Cyclotomic Fields....

    Hi everyone,

    How can I prove that Q(Zeta36) = Q(Zeta12,Zeta18) does not contain ANY primitive 5th roots of unity?

    [here, ZetaN = cos2pi/N + isin2pi/N ]

    Many many thanks in advance. x
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    Quote Originally Posted by AAM View Post
    Hi everyone,

    How can I prove that Q(Zeta36) = Q(Zeta12,Zeta18) does not contain ANY primitive 5th roots of unity?

    [here, ZetaN = cos2pi/N + isin2pi/N ]

    Many many thanks in advance. x
    In general \mathbb{Q}(\zeta_{\ell}) = \mathbb{Q}(\zeta_n,\zeta_m) where \ell = \text{lcm}(n,m).
    To prove this show each is a subset of eachother.
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  3. #3
    AAM
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    Hi perfecthacker - I think you may have misread my question - I wasn't asking about the equality of those 2 extensions - I was asking how to show that such an extension contains no primitive roots of unity.

    Many thanks. :-)
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    Quote Originally Posted by AAM View Post
    Hi perfecthacker - I think you may have misread my question - I wasn't asking about the equality of those 2 extensions - I was asking how to show that such an extension contains no primitive roots of unity.

    Many thanks. :-)
    Okay I fixed it.

    It can be shown that \mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m) = \mathbb{Q}(\zeta_d) where d=\gcd(n,m). Do you know this result? I can prove it for you.

    If \zeta_5\in \mathbb{Q}(\zeta_{36}) then \mathbb{Q}(\zeta_5)\subseteq \mathbb{Q}(\zeta_{36}) and so \mathbb{Q}(\zeta_5) \cap \mathbb{Q}(\zeta_{36}) = \mathbb{Q}(\zeta_5).
    But this is a problem because \gcd(36,5) = 1.

    By the way you can generalize this to prove that \mathbb{Q}(\zeta_n)\subseteq \mathbb{Q}(\zeta_m) if and only if n|m.
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  5. #5
    AAM
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    Beautiful! :-D

    Thank you SO much perfecthacker! :-)
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  6. #6
    AAM
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    One minor point though - you have done the question for Zeta_5 = cos2pi/5+isin2pi/5.

    But the original question was for Zeta = ANY PRIMITIVE 5th root of unity.

    So my question is does Q(Zeta) = Q(Zeta_5) ? Because if so then your argument still holds. :-)
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    Quote Originally Posted by AAM View Post
    Beautiful! :-D

    Thank you SO much perfecthacker! :-)
    Yes, I am beautiful.
    But I am all alone, , no girl wants to spend time with me.

    Quote Originally Posted by AAM View Post
    One minor point though - you have done the question for Zeta_5 = cos2pi/5+isin2pi/5.

    But the original question was for Zeta = ANY PRIMITIVE 5th root of unity.

    So my question is does Q(Zeta) = Q(Zeta_5) ? Because if so then your argument still holds. :-)
    Let \omega be a 5-th root of unity then the other primitive roots of unity are: \omega,\omega^2,\omega^3, \omega^4. Therefore, \zeta_5 \in \{ \omega,\omega^2,\omega^3,\omega^4\}. This means if \omega \in \mathbb{Q}_{36} then \{\omega,\omega^2,\omega^3,\omega^4\} \subset \mathbb{Q}_{36} and so \zeta_5 \in \mathbb{Q}_{36}. Now apply the argument above.

    This is Mine 12,5th Post!!!
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