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.