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Math Help - Cyclotomic Fields & Bases....

  1. #1
    AAM
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    Cyclotomic Fields & Bases....

    Hi everyone! :-)

    Suppose Zeta is any primitive 5th root of unity.

    Then we know that Q(Zeta) has a basis {1,Zeta,Zeta^2,Zeta^3} because Zeta is algebraic over Q.

    But how can I show that {Zeta,Zeta^-1,Zeta^2,Zeta^-2} is a basis? :-s

    Also, how does the Galois Group of Q(Zeta) act on this new basis?

    Furthermore, how can you prove the existence of a unique subfield E such that [Q(Zeta):E]=2, & find all a,b,c,d such that aZeta+bZeta^-1+cZeta^2+dZeta^-2 is in E but not Q? (apparently you can then show that E = Q(Sqrt(5)) !! :-o )

    I have absolutely no idea where to begin! :-s

    Any help on ANY of these questions would be amazing! :-)

    Thank you! x
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  2. #2
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    Quote Originally Posted by AAM View Post
    Hi everyone! :-)
    You like to smile a lot, .

    Suppose Zeta is any primitive 5th root of unity.

    Then we know that Q(Zeta) has a basis {1,Zeta,Zeta^2,Zeta^3} because Zeta is algebraic over Q.

    But how can I show that {Zeta,Zeta^-1,Zeta^2,Zeta^-2} is a basis? :-s
    Remember that \zeta^{-1} = \zeta^4 and \zeta^{-2} = \zeta^3.

    Also, how does the Galois Group of Q(Zeta) act on this new basis?
    The Galois group, G=\text{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q}) is isomorphic to the cyclic group \mathbb{Z}_4. Let \sigma : \mathbb{Q}(\zeta) \to \mathbb{Q}(\zeta) be the automorphism defined by \sigma (\zeta) = \zeta^2. Then G = \left< \sigma \right>. Since you know \sigma (\zeta) = \zeta^2 you can determine what \sigma (\zeta^k) is for k=2,3,4.

    Furthermore, how can you prove the existence of a unique subfield E such that [Q(Zeta):E]=2, & find all a,b,c,d such that aZeta+bZeta^-1+cZeta^2+dZeta^-2 is in E but not Q? (apparently you can then show that E = Q(Sqrt(5)) !! :-o )
    \zeta - \zeta^2 - \zeta^3 + \zeta^4= \sqrt{5}
    Therefore, E = \mathbb{Q}(\sqrt{5}) is a subfield of \mathbb{Q}(\zeta).
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