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

Abstract. Given an odd prime p we show a way to construct large families of polynomials Pq(x) ∈ Q[x], q ∈C,whereCis a set of primes of the form q ≡ 1modpand 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 ≥ 2andaprimeq≡1mod2n, 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)

Topics In Normal Bases Of Finite Fields

This is an introduction to the theory of normal bases of finite fields. The first few chapters cover a wide range of topics on the theory of normal bases of finite fields. Most standard definitions and results, including proofs are given. The last few chapters cover the theory of guassian and period normal bases of finite fields of low degrees. The last chapter presents the asymptotic proofs of the existence of primitive polynomials of degree n with approximately n/2 arbitrary coefficients, and primitive normal polynomials of arbitrary traces.Comment: 172 Pages; Keywords: Finite Fields, Guassian Period, Normal Bases, Primitive Normal Bases, Primitive Polynomial