Ovoids and primitive normal bases for quartic extensions of Galois fields

Abstract

We determine lower bounds for the number of primitive normal elements in a four-dimensional extension E over a Galois field $$F={\\mathrm{GF}}(q)$$\n . Our approach is based on viewing E as the three-dimensional projective space $$\\Gamma =\\mathrm{PG}(3,q)$$\n . In any of the three cases, whether q is even, or $$q\\equiv 3 { \\text{ mod } }4$$\n , or $$q\\equiv 1 { \\text{ mod } }4$$\n , we use a decomposition of the multiplicative group of E in order to determine a (canonical) partition of the point set of $$\\Gamma $$\n into $$q+1$$\n ovoids. The points of $$\\Gamma $$\n are distinguished into primitive and non-primitive ones, and an ovoid is called primitive if it contains at least one primitive point. The bounds are derived by studying the intersections of the primitive ovoids with the configuration of those points of $$\\Gamma $$\n which do not give rise to normal elements of E\xa0/\xa0F. Given that $$q^2+1$$\n is a prime number when q is even, or that $$\\frac{1}{2}(q^2+1)$$\n is a prime number when q is odd, we actually achieve the exact number of all primitive normal elements for the quartic extension over F.