Formula omitted]-valued trace of a finite-dimensional commutative [formula omitted]-algebra

Abstract

A non-zero F-valued F-linear map on a finite-dimensional commutative F-algebra is called an F-valued trace if its kernel contains no non-zero ideals. However, given an F-algebra, such a map may not always exist. We find an infinite class of finite-dimensional commutative F-algebras which admit an F-valued trace. In fact, in these cases, we explicitly construct a trace map. An F-valued trace on a finite-dimensional commutative F-algebra induces a non-degenerate bilinear form on the F-algebra which may be theoretically and computationally helpful. In this article, we suggest a couple of applications of an F-valued trace map of an F-algebra to algebraic coding theory.