ExplicitN-polynomials of 2-power degree over finite fields, I

Abstract

For an odd prime powerq the infinite field GF(q 2 ∞)= ⋃ n≥0 GF (q 2n ) is explicitly presented by a sequence (f n)≥1 ofN-polynomials. This means that, for a suitably chosen initial polynomialf 1, the defining polynomialsf n∈GF(q)[x] of degrees2 n are constructed by iteration of the transformation of variablex→x+1/x and have linearly independent roots over GF(q). In addition, the sequences are trace-compatible in the sense that the relative traces map the corresponding roots onto each other. In this first paper the caseq≡1 (mod 4) is considered and the caseq≡3 (mod 4) will be dealt with in a second paper. This specific construction solves a problem raised by A. Scheerhorn in [11].