Primitive normal bases for quartic and cubic extensions: a geometric approach

Abstract

For the Galois field extension $$\mathbb {F}_{q^n}$$Fqn over $$\mathbb {F}_q$$Fq we let $$PN_n(q)$$PNn(q) denote the number of primitive elements of $$\mathbb {F}_{q^n}$$Fqn which are normal over $$\mathbb {F}_q$$Fq. We derive lower bounds for $$PN_3(q)$$PN3(q) and $$PN_4(q)$$PN4(q), the number of primitive normal elements in cubic and quartic extensions. Our reasoning relies on basic projective geometry. A comparision with the exact numbers for $$PN_3(q)$$PN3(q) and $$PN_4(q)$$PN4(q) where $$q\le 32$$q≤32 (altogether 36 instances) indicates that these bounds are very good; we even achieve equality in 14 cases.