The strong primitive normal basis theorem

Abstract

An element α of the extension E of degree n over the finite field F = GF (q) is called free over F if {α, α q , . . . , α q n−1 } is a (normal) basis of E/F . The primitive normal basis theorem, first established in full by Lenstra and Schoof (1987), asserts that for any such extension E/F , there exists an element α ∈ E such that α is simultaneously primitive (i.e., generates the multiplicative group of E) and free over F . In this paper we prove the following strengthening of this theorem: aside from five specific extensions E/F , there exists an element α ∈ E such that both α and α −1 are simultaneously primitive and free over F .

An extension of the (strong) primitive normal basis theorem

“…Following Cohen and Huczynska [8,9], we introduce a sieve that will help us relax the condition proved in the previous section. The propositions included in this section are those of Cohen and Huczynska [9], adjusted properly.…”

“…It can be shown [8,9] that Ω F is the characteristic function for the elements of F q m that are F-free over F q .…”

“…Following Cohen and Huczynska [8,9], we define the characteristic function of the r -free elements of F q m as follows:…”

“…There exists some x ∈ F q m such that x and x −1 are both simultaneously primitive and free over F q , unless the pair (q, m) is one of (2,3), (2,4), (3,4), (4,3) or (5,4). Tian and Qi [26] were the first to prove this result for m ≥ 32, but Cohen and Huczynska [9] were those who extended it to its stated form, once again with the help of their sieving techniques. The reader is referred to [6,17] and the references therein, for complete surveys of this, very active, line of research.…”

“…Moreover, much of this paper is inspired by the work of Cohen and Huczynska [8,9], whose sieving techniques have been adjusted and partially implemented. All non-trivial calculations were performed with MAPLE (v. 13).…”