On the Product of Elements with Prescribed Trace

Abstract

© 2020 Australian Mathematical Publishing Association Inc. This paper deals with the following problem. Given a finite extension of fields and denoting the trace map from to by, for which elements in, and, in, is it possible to write as a product, where with ? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least 5. We also have results for arbitrary fields and extensions of degrees 2, 3 or 4. We then apply our results to the study of perfect nonlinear functions, semifields, irreducible polynomials with prescribed coefficients, and a problem from finite geometry concerning the existence of certain disjoint linear sets.