On Density of Primitive Elements for Field Extensions

This paper presents an explicit bound on the number of primitive elements that are linear combinations of generators for field extensions. It is well known that every finite separable extension of an arbitrary field has a primitive element; that is, there exists a single element in the extension field which generates that field over the ground field. This is a fundamental theorem in algebra which is called the primitive element theorem in many textbooks, see for example [2, 5, 6], and it is a useful tool in practical computation of commutative algebra [4]. The existence proofs found in the literature make no attempt at estimating the density of primitive elements. The purpose of this paper is to give an explicit lower bound on the density of primitive elements that are linear combinations of generators. Our derivation uses a blend of Galois theory, basic linear algebra, and a simple form of the principle of inclusion-exclusion from elementary combinatorics. Additionally, for readers familiar with Grobner bases, we show by a geometric example, how to test a linear combination for primitivity without relying on the Galois groups used in deriving our bound. Let F be any field and K a finite algebraic extension of F. An element β ∈ K is called primitive for K over F if K = F(β). Suppose K is generated by α1, . . . , αn, that is, K = F(α1, . . . , αn). Consider elements of the form β = b1α1 + · · ·+ bnαn where bi ∈ F, 1 ≤ i ≤ n. We would like to know when and how frequently such elements are primitive for K over F when the coefficients are required to come from Date: August 10, 2004. 1991 Mathematics Subject Classification. Primary 12Y05; Secondary 12F10.