SPECIFIC IRREDUCIBLE POLYNOMIALS WITH LINEARLY INDEPENDENT ROOTS OVERFINITE-FIELDS

We give several families of specific irreducible polynomials with the following property: if f(x) is one of the given polynomials of degree n over a finite field F-q and alpha is a root of it, then alpha epsilo n F-qn is normal over every intermediate field between F-qn and F-q. H ere by alpha epsilon F-qn being normal over a subfield F-q we mean tha t the algebraic conjugates alpha, alpha(q),..., alpha(qn-1) are linear ly independent over F-q. The degrees of the given polynomials are of t he form 2(k) or Pi(i=1)(u) r(i)(li) where r(1), r(2),..., r(u) are dis tinct odd prime factors of q - 1 and k, l(1),..., l(u) are arbitrary p ositive integers. For example, we prove that, for a prime p = 3 mod 4, if x(2) - bx - 1 epsilon F-p[x] is irreducible with b not equal 2 the n the polynomial (x - 1)(2k+1) - b(x - 1)(2k)x(2k) - x(2k+1) has the d escribed property over F-p for every integer k greater than or equal t o 0. We also show how to efficiently compute the required b epsilon F- p. (C) Elsevier Science Inc., 1997.