Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers

Given an odd prime p we show a way to construct large families of polynomials Pq(x) ∈ Q[x], q ∈ C, where C is a set of primes of the form q ≡ 1 mod p and Pq(x) is the irreducible polynomial of the Gaussian periods of degree p in Q(ζq). Examples of these families when p = 7 are worked in detail. We also show, given an integer n ≥ 2 and a prime q ≡ 1 mod 2n, how to represent by matrices the Gaussian periods η0, . . . , ηn−1 of degree n in Q(ζq), and how to calculate in a simple way, with the help of a computer, irreducible polynomials for elements of Q(η0). Introduction Let p be an odd prime number and ζp a p-th primitive root of 1. Let S be the set of all primes q ≡ 1 mod p. For each q ∈ S, choose a primitive root s = sq modulo q, and a q-th primitive root ζq of 1 (the choice of sq will be made more precise later). For q ∈ S and 0 ≤ i ≤ p − 1, define the Gaussian periods of degree p in Q(ζq) by ηi = ηi(q) = ∑f−1 j=0 ζ si+pj q , where f = fq = (q − 1)/p. In Section 1 we show a way to construct large families of polynomials Pq(x) ∈ Q[x], q ∈ C, where C ⊆ S and Pq(x) is the irreducible polynomial of the periods ηi(q). When p is small, we could take C = S. More precisely, for any p, the set of indices could, in principle, include all primes q ∈ S such that the prime ideals over q in Q(ζp) are principal. Of course, if we do not put some restrictions, the formulas describing these families will soon become enormously complicated. As examples we show, for p = 7, four two-parameter families of polynomials Pq(x), whose indices put together include all the primes of the form (a + b)/(a + b) (see Proposition 1, and the MAPLE program, at the end of Section 1, to calculate the polynomials Pq(x) for the four families). These results can be generalized to arbitrary positive integers in the place of the primes p. For p = 5, H.W. Lloyd Tanner obtained, in [9], an expression for the family of polynomials Pq(x), q ∈ S, in terms of coefficients of certain divisors of q in Q(ζ5). This result was used by Emma Lehmer, in [5], who gave a new expression for that family. In [6] Lehmer shows a family of polynomials of degree 5, which is obtained by a translation of a family of polynomials Pq(x), and such that the roots of the polynomials in the family are units. This result has been used by Schoof and Washington in [7] to find some real cyclotomic fields with large class numbers. Received by the editor May 19, 1998 and, in revised form, October 15, 1998. 1991 Mathematics Subject Classification. Primary 11R18, 11R21, 11T22. This work was supported in part by grants from NSERC and FCAR. c ©2000 American Mathematical Society 1653