Stephen D. Cohen

Primitive elements and polynomials with arbitrary trace

Abstract With one non-trivial exception, GF(qn) contains a primitive element of arbitrary trace over GF(q).

A class of exceptional polynomials

We present a class of indecomposable polynomials of non primepower degree over the finite field of two elements which are permutation polynomials on infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple groups where k ≥ 3 and odd. (The first member of this class was previously found by P. Muller [17]. This realises one of only two possibilities for such a class which remain following deep work of Fried, Guralnick and Saxl [7]. The other is associated with, psl(3) k ≥ 3, and odd in fields of characteristic 3.

The explicit construction of irreducible polynomials over finite fields

For a finite field GF(q) of odd prime power order q, and n ≥ 1, we construct explicitly a sequence of monic irreducible reciprocal polynomials of degree n2m (m = 1, 2, 3, ...) over GF(q). It is the analog for fields of odd order of constructions of Wiedemann and of Meyn over GF(2). We also deduce iterated presentations of GF (qn2∞).

Commutative semifields, two dimensional over their middle nuclei

Then we say that S is a (finite) semzj?eld. The subset N, = (n E S: (xn)y = x(ny), Vx, y E S} is known as the middle nucleus of S. N, is a field, and S may be regarded as a (left or right) vector space over N,. A semifield in which multiplication is not associative (i.e., N, # S) is called proper. A finite semifield necessarily has prime power order and proper finite semifields exist for all orders q = p’, p prime, where Y > 3 if p is odd and r>4ifp=2. There is a considerable link between finite semifields and finite projective planes of Lenz-Barlotti class V.1. Indeed, any finite proper semilield gives rise to such a plane, and conversely any plane of the above type yields many isotopic, but not necessarily isomorphic, semifields. The paper by Knuth [4] is an excellent survey of finite semifields and their connections with projective planes. We mention one of the examples given in 141‘

Primitive Normal Bases with Prescribed Trace

Abstract. Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let a∈F be nonzero. We prove the existence of an element w in E satisfying the following conditions: - w is primitive in E, i.e., w generates the multiplicative group of E (as a module over the ring of integers). - the set {wg∣g∈G} of conjugates of w under G forms a normal basis of E over F. - the (E, F)-trace of w is equal to a.This result is a strengthening of the primitive normal basis theorem of Lenstra and Schoof [10] and the theorem of Cohen on primitive elements with prescribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of such elements for the cases in which q≤ 97 and n≤ 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically.

Explicit theorems on generator polynomials

Progress over the past decade is surveyed concerning explicit existence and construction theorems on irreducible, primitive and normal polynomials.

The Primitive Normal Basis Theorem – Without a Computer

Given

Primitive polynomials with a prescribed coefficient

q

Primitivity, freeness, norm and trace

, a power of a prime

Primitive elements in finite fields and costas arrays

p