A Characterization of Primitive Polynomials over Finite Fields

AbstractWe characterize primitive polynomials, among irreducible polynomials, by the number of terms in a certain quotient. We apply this to BCH codes of maximal designed distance

Norms of Sums of Squares

For a finite separable extension K/F of fields of characteristic not 2, the norm of a sum of 2n squares in K is a sum of 2n squares in F. We find explicit identities

Octonions

We present the basic properties of the octonions and construct the five exceptional simple Lie algebras

Invariants of Trace Forms over Finite Fields of Characteristic 2

AbstractLet K be a finite extension of F2. We compute the invariants of the quadratic form Q(x)=trK/F2(x(x2a+x2b)) and so determine the number of zeros in K. This is applied to finding the cross-correlation of certain binary sequences

Highly Degenerate Quadratic Forms over Finite Fields of Characteristic 2

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

Norm Euclidean Quaternionic Orders

We determine the norm Euclidean orders in a positive definite quaternion algebra over Q

Torsion-Free Modules over Reduced Witt Rings

We compute the genus class group of a torsion-free module over a reduced Witt ring of finite stability index. This is applied to modules locally isomorphic to odd degree extensions of formally real fields

Gorenstein Witt Rings II

The abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classify the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type

Trace Forms over Finite Fields of Characteristic 2 with Prescribed Invariants

AbstractWe determine the possible pairs of invariants of a trace form and construct forms with these invariants. We use this to construct new maximal Artin–Schreier curves