Ovoids and primitive normal bases for quartic extensions of Galois fields

We determine lower bounds for the number of primitive normal elements in a four-dimensional extension E over a Galois field F=GF(q). Our approach is based on viewing E as the three-dimensional projective space Gamma=PG(3,q). In any of the three cases, whether q is even, or q equivalent to 3mod4, we use a decomposition of the multiplicative group of E in order to determine a (canonical) partition of the point set of Gamma 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 is a prime number when q is even, or that 12(q2+1) 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.