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Thread: 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? $\displaystyle \Phi_n$ is the n-th cyclotomic polynomial.

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

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

    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? $\displaystyle \Phi_n$ is the n-th cyclotomic polynomial.

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

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

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

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

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

    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 $\displaystyle \Longleftarrow$ of 2, namely:
    if $\displaystyle p\equiv 1 \pmod n$ then $\displaystyle p\mid \Phi_n(a)$ for some $\displaystyle a\in\mathbb{Z}$.

    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 $\displaystyle \Longleftarrow$ of 2, namely:
    if $\displaystyle p\equiv 1 \pmod n$ then $\displaystyle p\mid \Phi_n(a)$ for some $\displaystyle a\in\mathbb{Z}$.

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