Jacobi sums and new families of irreducible polynomials of Gaussian periods

Let m > 2, ζm an m-th primitive root of 1, q ≡ 1 mod 2m a prime number, s = sq a primitive root modulo q and f = fq = (q − 1)/m. We study the Jacobi sums Ja,b = − ∑q−1 k=2 ζ a inds(k)+b inds(1−k) m , 0 ≤ a, b ≤ m−1, where inds(k) is the least nonnegative integer such that s inds(k) ≡ 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 Pq(x), q ∈ P, of irreducible polynomials of Gaussian periods, ηi = ∑f−1 j=0 ζ si+mj q , of degree m, where P is a suitable set of primes ≡ 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. Introduction Let m > 2 be an integer and ζm an m-th primitive root of 1. For each prime q ≡ 1 mod 2m let ζq be a q-th primitive root of 1, s = sq a primitive root modulo q and f = fq = (q − 1)/m (we will assume that f is even for simplicity). Let S be the set of all primes q ≡ 1 mod 2m. Given q ∈ S, define the Jacobi sums Ja,b, 0 ≤ a, b ≤ m−1, and the Gaussian periods ηi, 0 ≤ i ≤ m−1, of degree m in Q(ζq), by Ja,b = − q−1 ∑ k=2 ζ a inds(k)+b inds(1−k) m , where inds(k) is the least nonnegative integer such that s inds(k) ≡ k mod q, and ηi = f−1 ∑ j=0 ζ i+mj q . Define Pq(x) = ∏m−1 i=0 (x−ηi), the irreducible polynomial, over Q, of the periods ηi. In this article we study the numbers Ja,b, and use them to construct large families of polynomials Pq(x), q ∈ P , where P is a subset of S. In principle the method shown here would allow us to construct a finite number of such families, whose indices put together include all the primes in S. This research originated from a problem indicated to me by René Schoof. The first part of the problem was to find, for m = 7, or m = 9, or m = 12, families of Received by the editor September 15, 1998 and, in revised form, January 19, 2000. 2000 Mathematics Subject Classification. Primary 11R18, 11R21, 11T22. This work was supported in part by grants from NSERC and FCAR. c ©2001 American Mathematical Society 1617