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Math Help - cyclotomic polynomials

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

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

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

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

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

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

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

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

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

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