For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF (2) in which the coefficients of x n−1 , x n−2 and x n−3 are specified in advance. Formulas for the number of elements in GF (2 n) with the first three traces specified are also given.

Abstract. We give, over a finite field F q , explicit factorizations into a product of irreducible polynomials, of the cyclotomic polynomials of order 3 · 2 n , the Dickson polynomials of the first kind of order 3 · 2 n and the Dickson polynomials of the second kind of order 3 · 2 n − 1.

Let K/F be an extension of finite fields of characteristic two. We consider quadratic forms written as the trace of xR(x), where R(x) is a linearized polynomial. We show all quadratic forms can be so written, in an essentially unique way. We classify those R, with coefficients 0 or 1, where the form has a codimension 2 radical. This is applied to maximal Artin-Schreier curves and factorizations of linearized polynomials. v i=1 x i y i. Here s is any element of F with tr F/GF (2) (s) = 1.

We develop a matrix approach to compute a certain sum of Gauss sums which arises in the study of weights of irreducible codes. A lower bound on the minimum weight of certain irreducible codes is given.

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