Specific irreducible polynomials with linearly independent roots over finite fields Original Research Article

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 Fq and α is a root of it, then α set membership, variant Fqn is normal over every intermediate field between Fqn and Fq. Here by α set membership, variant Fqn being normal over a subfield Fq we mean that the algebraic conjugates α, αq, …, αqn−1 are linearly independent over Fq. The degrees of the given polynomials are of the form 2k or Πui=1 rlii where r1, r2, …, ru are distinct odd prime factors of q − 1 and k, l1, …, lu are arbitrary positive integers. For example, we prove that, for a prime p ≡ 3 mod 4, if x2 − bx − 1 set membership, variant Fp[x] is irreducible with b = 2 then the polynomial (x − 1)2k+1 - b(x − 1)2kx2k − x2k+1 has the described property over Fp for every integer k greater-or-equal, slanted 0. We also show how to efficiently compute the required b set membership, variant Fp.