Open problems and conjectures in finite fields

Abstract

Abstract We discuss a number of open problems and conjectures in the theory and application of finite fields. We also provide a brief discussion of the status as well as references related to each problem. Introduction In this paper we try to summarise some interesting and/or important questions in the theory and application of finite fields. These questions obviously reflect our personal tastes but we have indeed tried to consider questions of general interest. We hope that these questions and even more, the methods developed for their solutions, will be of interest to other researchers. The reader may wonder why some questions we call ‘Problems’ and some we call ‘Conjectures’. Roughly speaking, we use the term Conjecture if we (and very often others) believe the statement to be true while the term Problem is used to indicate all other statements. In general we feel that our conjectures may be more difficult to resolve than our problems. The standard notions of the theory of finite fields which we use can be found in the finite field bible by Lidl and Niederreiter. Combinatorics There are numerous open problems in combinatorics which are related to finite fields. In this section we briefly describe several of these. We begin with several questions related to latin squares. A latin square of order n is an n × n array based upon n distinct symbols with the property that each row and each column contains each of the n symbols exactly once.