ABSTRACT
Certain kinds of rings come up so often that they are given special names. Among the most important of these kinds of rings are the fields. A field is a commutative ring in which every nonzero element is a unit. This chapter shows how to construct finite fields and what their arithmetic is like. Euclid’s proof shows that there are an infinite number of irreducible polynomials over any field. Over a finite field, there are irreducible polynomials of each positive degree. Over the complex numbers the only irreducible polynomials are the linear ones. That’s one way of stating the fundamental theorem of algebra. The chapter describes Kronecker’s construction of simple field extensions by using a four-element field and a sixteen-element field. It also provides a theorem proving that any two finite fields with the same number of elements are isomorphic.
