Completely normal elements in iterated quadratic extensions of finite fields
Given a finite field \(F_q\) and an integer \(n \geq 1\), an element \(\alpha\) of \(F_{q^n}\) is called normal over \(F_q\) if its \(F_q\)-conjugates form an \(F_q\)-vector space basis of \(F_{q^n}\). The element \(\alpha\) is called \textit{completely normal over \(F_q\)}, if, simultaneously, for each divisor \(d\) of \(n\), the \(F_{q^d}\)-conjugates of \(\alpha\) constitute an \(F_{q^d}\)-vector space basis of \(F_{q^n}\). For certain ground fields \(F_q\) and certain degrees \(n\) the author provides explicit constructions of (sequences of) irreducible polynomials whose roots are completely normal elements. More precisely, the results are as follows: For \(q \equiv 1\) mod \(4\) and \(n\) a power of \(2\) it is shown that \textit{S. D. Cohen}'s recursive construction of irreducible polynomials of degree \(n\) over \(F_q\) [Des. Codes Cryptography 2, 169-174 (1992; Zbl 0768.11048)] yields polynomials whose roots are completely normal over \(F_q\). The normality of these elements was previously proved by \textit{H. Meyn} [Des. Codes Cryptography 6, 107-116 (1995; Zbl 0834.11053)]. For \(q \equiv 3\) mod \(4\) and \(n\) a power of \(2\) it is shown that McNay's recursive construction of irreducible polynomials of degree \(n\) over \(F_q\) [\textit{G. McNay}, ``Topics in Finite Fields'', Ph.D. thesis, University of Glasgow (1995)] yields polynomials whose roots are completely normal over \(F_q\). The author also discusses a generalization of the polynomial transformations of Cohen and McNay in order to obtain sequences of irreducible polynomials of degree a power of an odd prime \(r\) for ground fields \(F_q\) where \(q \equiv 1\bmod 2r\). Again the roots of these polynomials are completely normal over \(F_q\).
scientific article; zbMATH DE number 1199983
scientific article