is studied. Simple criteria are given for the case that the irreducibility of f is inherited by the self-reciprocal polynomial fQ. Infinite sequences of irreducible self-reciprocal polynomials are constructed by iteration of this Q-transformation.
The aim of this note is to show that the (well-known) factorization of the 2 nϩ1 th cyclotomic polynomial x 2 n ϩ 1 over GF(q) with q ϵ 1 (mod 4) can be used to prove the (more complicated) factorization of this polynomial over GF(q) with q ϵ 3 (mod 4).
Abstract.For an odd prime power q the infinite field GF(q 2~ ) = U,_>0 GF(q2~ ) is explicitly presented by a sequence (f,,),,_>l of N-polynomials. This means that, for a suitably chosen initial polynomial fl, the defining polynomials f, ~ GF(q ) [x ] of degrees 2 n are constructed by iteration of the transformation of variable x ~+ 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 case q ---1 (mod 4) is considered and the case q ---3 (rood 4) will be dealt with in a second paper. This specific construction solves a problem raised by A. Scheerhorn in [11].
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