Construction of projective planes

Abstract

Abstract In this chapter we turn our attention to rings with only a finite number of elements, and in particular to finite fields. (Recall that a field is a commutative ring with identity 1≠0 in which each non-zero element has a multiplicative inverse.) Finite fields play an important role in many branches of mathematics. We begin by showing how the existence of certain finite fields leads, via projective geometry, to the solution of a particular problem in combinatorial mathematics. We then go on to investigate the existence of finite fields. Several of the concepts that we introduce in connection with finite fields will be of use when we later turn our attention to infinite fields. For this reason we give definitions in more generality than at first glance seems necessary.