The form of a cyclotomic polynomial

Question

If $n$ is a prime number then the cyclotomic polynomial $\Phi_n(x)$ has the form $\sum_{k=0}^{n-1}x^k$.

Is the converse also true, i.e. has $\Phi_n(x)$ this form only if $n$ is prime?

Answer 1

If $n>1$, the cyclotomic polynomial $\Phi_n(x)$ has degree $\phi(n)$, which is always less than $n-1$, unless $n$ is prime.

Hence, if $n$ is composite,$\Phi_n(x)$ can't have the form $x^{n-1} + \cdots + 1$.