Specific irreducible polynomials with linearly independent roots over finite 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 epsilon F-qn is normal over every intermediate field between F-qn and F-q. Here by alpha epsilon F-qn being normal over a subfield F-q we mean that the algebraic conjugates alpha, alpha(q),..., alpha(qn-1) are linearly independent over F-q. The degrees of the given polynomials are of the form 2(k) or Pi(i=1)(u) r(i)(li) where r(1), r(2),..., r(u) are distinct odd prime factors of q - 1 and k, l(1),..., l(u) are arbitrary positive 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 then the polynomial (x - 1)(2k+1) - b(x - 1)(2k)x(2k) - x(2k+1) has the described property over F-p for every integer k greater than or equal to 0. We also show how to efficiently compute the required b epsilon F-p. (C) Elsevier Science Inc., 1997.