Explicit Factorization of Some Period Polynomials

Abstract

Let p, t, q, n, m and r be positive integers, such that p is a prime number, $q=p^t$, $\gcd (q,n)=1$, $m=\text{ ord}_n(q)$, and suppose that the prime factors of r divide n but not $(q^m-1)/n$, and that $q^m \equiv 1 \pmod {4}$, if 4|r. Also let u such that $u=\gcd (\frac{q^m-1}{q-1},\frac{q^m-1}{n})$. Assume that $u=1$ or p is semiprimitive modulo u. Under these conditions, we are going to obtain the explicit factorization of the period polynomial of degree $\gcd (\frac{q^{mr}-1}{q-1},\frac{q^{mr}-1}{nr})$ for the finite field ${\mathbb {F}}_{q^{mr}}$. In fact, we will see that such polynomial has always integer roots, meaning that the corresponding Gaussian periods are also integer numbers. As an application, we also determine the number of solutions of certain diagonal equations with constant exponent.

Partially supported by PAPIIT-UNAM IN109818.