One small problem with cyclotomic polynomials

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April 24th 2010, 07:21 AM
AMA

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.
April 24th 2010, 03:05 PM
NonCommAlg

Quote:

Originally Posted by AMA

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.
April 26th 2010, 06:07 AM
AMA

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.
April 27th 2010, 07:27 PM
FancyMouse

Quote:

Originally Posted by AMA

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 $(\mathbb{Z}/p\mathbb{Z})^*$ ?