Irreducible polynomials with prescribed trace and restricted norm

Details Irreducible polynomials with prescribed trace and restricted norm Irreducible polynomials with prescribed trace and restricted norm

2007-10-15

arxiv.org/abs/0710.2767v1

Mathematics

Number Theory

25 pages

Scientific paper

Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a coset of a subgroup , s|(q-1). We give the number explicitly for s=2, 3, 4 when q=p, and for s|(p^e+1) when r=2en. Finally, we give explicit formulae for the number when both trace and norm are fixed, p=2 and m =< 30.

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