Jacobi sums and new families of irreducible polynomials of Gaussian periods

Abstract:

Let $m> 2$, $\zeta _m$ an $m$-th primitive root of 1, $q\equiv 1$ mod $2m$ a prime number, $s=s_{q}$ a primitive root modulo $q$ and $f=f_{q}=(q-1)/m$. We study the Jacobi sums $J_{a,b}=-\sum _{k=2}^{q-1}\zeta _m ^{ a \text {ind}_{s}(k)+b \text {ind}_{s}(1-k)}$, $0\leq a, b\leq m-1$, where $\text {ind}_{s}(k)$ is the least nonnegative integer such that $s^{ \text {ind}_{s}(k)}\equiv k$ mod $q$. We exhibit a set of properties that characterize these sums, some congruences they satisfy, and a MAPLE program to calculate them. Then we use those results to show how one can construct families $P_{q}(x)$, $q\in \mathcal {P}$, of irreducible polynomials of Gaussian periods, $\eta _{i}=\sum _{j=0}^{f-1}\zeta _q^{s^{i+mj}}$, of degree $m$, where $\mathcal {P}$ is a suitable set of primes $\equiv 1$ mod $2m$. We exhibit examples of such families for several small values of $m$, and give a MAPLE program to construct more of them.