# Thread: One small problem with cyclotomic polynomials

1. ## 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:
Page 324, problem 21.

Thanks a lot.

2. Originally Posted by AMA
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:
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.

3. Hello again,

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.

4. Originally Posted by AMA
Hello again,

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