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Math Help - One small problem with cyclotomic polynomials

  1. #1
    AMA
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    One small problem with cyclotomic polynomials

    Hello,

    Can someone help me solve this problem with cyclotomic polynomials? \Phi_n is the n-th cyclotomic polynomial.

    1. Let a be a non-zero integer, p a prime, n a positive integer and p\nmid n. Prove that p\mid \Phi_n(a) if and only if a has period n in (\mathbb{Z}/p\mathbb{Z})^*.

    2. Again assume p\nmid n. Prove that p\mid \Phi_n(a) for some a\in\mathbb{Z} if and only if p\equiv 1 \pmod n

    Here's the source of the problem:
    Algebra - Google Livres
    Page 324, problem 21.

    Thanks a lot.
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  2. #2
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    Quote Originally Posted by AMA View Post
    Hello,

    Can someone help me solve this problem with cyclotomic polynomials? \Phi_n is the n-th cyclotomic polynomial.

    1. Let a be a non-zero integer, p a prime, n a positive integer and p\nmid n. Prove that p\mid \Phi_n(a) if and only if a has period n in (\mathbb{Z}/p\mathbb{Z})^*.

    2. Again assume p\nmid n. Prove that p\mid \Phi_n(a) for some a\in\mathbb{Z} if and only if p\equiv 1 \pmod n

    Here's the source of the problem:
    Algebra - Google Livres
    Page 324, problem 21.

    Thanks a lot.
    let f(x)=x^n-1=\prod_{d \mid n} \Phi_d(x). see that if t is the order of a modulo p, then p \mid \Phi_t(a). now suppose that p \mid \Phi_n(a). then p \mid a^n - 1 and thus t \mid n. that means, in (\mathbb{Z}/p\mathbb{Z})[x],

    both \Phi_n(x) and \Phi_t(x) are divisible by x-a and so if t \neq n, then (x-a)^2 \mid f(x). hence x - a \mid f'(x)=nx^{n-1}, which gives us na^{n-1} \equiv 0 \mod p and so p \mid n.

    do the rest of the problem yourself.
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  3. #3
    AMA
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    Hello again,

    Thanks for your help, NonCommAlg!

    I just can't find a way to proof the \Longleftarrow of 2, namely:
    if p\equiv 1 \pmod n then p\mid \Phi_n(a) for some a\in\mathbb{Z}.

    Thanks.
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  4. #4
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    Quote Originally Posted by AMA View Post
    Hello again,

    Thanks for your help, NonCommAlg!

    I just can't find a way to proof the \Longleftarrow of 2, namely:
    if p\equiv 1 \pmod n then p\mid \Phi_n(a) for some a\in\mathbb{Z}.

    Thanks.
    Then what happens in (\mathbb{Z}/p\mathbb{Z})^*?
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