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Thread: cyclotomic polynomials

  1. #1
    Senior Member Sampras's Avatar
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    cyclotomic polynomials

    We know that $\displaystyle \Phi_{p}(x) = \frac{x^p-1}{x-1} = x^{p-1}+x^{p-2}+ \dots + x+1 $ is irreducible over $\displaystyle \mathbb{Q} $ for every prime $\displaystyle p $. Suppose $\displaystyle \gamma $ is a zero of $\displaystyle \Phi_{p}(x) $ and consider the field $\displaystyle \mathbb{Q}(\gamma) $.

    (a) Show that $\displaystyle \gamma, \gamma^2, \dots, \gamma^{p-1} $ are distinct zeros of $\displaystyle \Phi_{p}(x) $ and show that they are all the zeros of $\displaystyle \Phi_{p}(x) $.

    (b) Show that $\displaystyle G(Q(\gamma)/ \mathbb{Q}) $ is abelian of order $\displaystyle p-1 $.

    (c) Show that the fixed field of $\displaystyle G(\mathbb{Q}(\gamma)/ \mathbb{Q}) $ is $\displaystyle \mathbb{Q} $.

    For (a), these are the nth roots of unity? For (b), consider two automorphisms of $\displaystyle \mathbb{Q}(\gamma) $ and show that they commute? For (c), isn't this by definition?
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    Quote Originally Posted by Sampras View Post
    We know that $\displaystyle \Phi_{p}(x) = \frac{x^p-1}{x-1} = x^{p-1}+x^{p-2}+ \dots + x+1 $ is irreducible over $\displaystyle \mathbb{Q} $ for every prime $\displaystyle p $. Suppose $\displaystyle \gamma $ is a zero of $\displaystyle \Phi_{p}(x) $ and consider the field $\displaystyle \mathbb{Q}(\gamma) $.

    (a) Show that $\displaystyle \gamma, \gamma^2, \dots, \gamma^{p-1} $ are distinct zeros of $\displaystyle \Phi_{p}(x) $ and show that they are all the zeros of $\displaystyle \Phi_{p}(x) $.
    First, the zero of $\displaystyle \Phi{p}(x)$ are just the zeros of $\displaystyle x^p- 1$, excluding 1. If if $\displaystyle \gamma$ is such a 0, then $\displaystyle \gamma^p= 1$ so that $\displaystyle (\gamma^n)^p= (\gamma^p)^n= 1$. The only 'hard' part is showin that $\displaystyle \gamma^n$ is not 1 for n= 1, 2, ..., p-1. It is important that p is prime. here.
    Suppose $\displaystyle \gamma^i= \gamma^j$ for i< j< p. Then $\displaystyle \gamma^{j- i}= 1$. Show that that is impossible.

    [tex](b) Show that $\displaystyle G(Q(\gamma)/ \mathbb{Q}) $ is abelian of order $\displaystyle p-1 $.

    (c) Show that the fixed field of $\displaystyle G(\mathbb{Q}(\gamma)/ \mathbb{Q}) $ is $\displaystyle \mathbb{Q} $.

    For (a), these are the nth roots of unity? For (b), consider two automorphisms of $\displaystyle \mathbb{Q}(\gamma) $ and show that they commute? For (c), isn't this by definition?[/QUOTE]
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