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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?

Sophia Antipolis
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    Yes. If $n=ab$, then $\sum_{k=0}^{n-1} x^k$ factors as $\big(\sum_{k=0}^{a-1}x^k\big)\cdot \big(\sum_{k=0}^{b-1}x^{ak}\big)$, but cyclotomic polynomials are irreducible: so if $n$ is prime, $\Phi_n$ is of this form; if $n$ is not prime, then $\Phi_n$ is not of this form. – anthonyquas Mar 05 '18 at 07:13

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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 $x^{n-1} + \cdots + 1$.

quasi
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