Jacobi sums and new families of irreducible polynomials of Gaussian periods

Abstract

Abstract. 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 sumsWe 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,, 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.

Class numbers of real cyclotomic fields of prime conductor

“…The class numbers of Emma Lehmer's degrees 5 and 8 have been computed in [17], [20] for all l < 10 10 . There are no similar families known for degree 7, 9 or larger, but see [21]. In the range of our computations we encountered several members of these families of number fields.…”

Cyclic polynomials and the multiplication matrices of their roots

“…Finally we tie that information together to give a method to construct the mentioned cyclic polynomials and matrices A (Theorem 2). This method is more general than the one given in [10] both because we are working now over arbitrary integrally closed characteristic zero domains D (and not only over Z[n 1 ; : : : ; n r ]) and because we ÿnd generalizations of irreducible polynomials of real and complex Gaussian periods (not only of real Gaussian periods). As applications we construct families of polynomials with cyclic and dihedral Galois groups over Q and of polynomials with cyclic Galois groups over quadratic ÿelds .…”

“…In the following proposition we list some known properties of Jacobi sums (the list appears in [10,Proposition 2], for the case f even). We will show later other elements which also satisfy these properties and will use them to construct new cyclic polynomials and the corresponding multiplication matrices.…”

Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations

“…This fact is applied also to other one-parameter families of cyclic polynomials obtained from polynomials of Gaussian periods in [9], [19], [20]. Based on this observation, one can expect a new method of systematic construction of cyclic polynomials, which is essentially different from those described in the literature of the inverse Galois problem, such as [15], [11].…”

“…This method of constructing units of cyclic extensions of rationals is known as the "Lehmer project" [8]. Recently, Thaine [19,20] made an extensive study of irreducible polynomials of Gaussian periods of arbitrary degree e and constructed new one-parameter families of cyclic polynomials explicitly for degree 7, 9 and 12. His method, which uses cyclotomic numbers, Jacobi sums, and Stickelberger's theorem for a rational prime which is the norm of a certain number α ∈ Z[ζ e ], seems to be very interesting from the viewpoint of the Lehmer project.…”