Jacobi sums and new families of irreducible polynomials of Gaussian periods

Abstract

Let m > 2 m> 2 , ζ m \zeta _m an m m -th primitive root of 1, q ≡ 1 q\equiv 1 mod 2 m 2m a prime number, s = s q s=s_{q} a primitive root modulo q q and f = f q = ( q − 1 ) / m f=f_{q}=(q-1)/m . We study the Jacobi sums J a , b = − ∑ k = 2 q − 1 ζ m a ind s ( k ) + b ind s ( 1 − k ) J_{a,b}=-\sum _{k=2}^{q-1}\zeta _m ^{\, a\, \text {ind}_{s}(k)+b\, \text {ind}_{s}(1-k)} , 0 ≤ a , b ≤ m − 1 0\leq a, b\leq m-1 , where ind s ( k ) \text {ind}_{s}(k) is the least nonnegative integer such that s ind s ( k ) ≡ k s^{\, \text {ind}_{s}(k)}\equiv k mod q 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 ) P_{q}(x) , q ∈ P q\in \mathcal {P} , of irreducible polynomials of Gaussian periods, η i = ∑ j = 0 f − 1 ζ q s i + m j \eta _{i}=\sum _{j=0}^{f-1}\zeta _q^{s^{i+mj}} , of degree m m , where P \mathcal {P} is a suitable set of primes ≡ 1 \equiv 1 mod 2 m 2m . We exhibit examples of such families for several small values of m m , and give a MAPLE program to construct more of them.