Galois Theory, 5th Edition

Chapter 19Finite Fields

DOI: 10.1201/9781003213949-19

Fields that have finitely many elements are important in many branches of mathematics, including number theory, group theory, and projective geometry. They also have practical applications, especially to the coding of digital communications, see Lidl and Niederreiter (1986), and, especially for the history, Thompson (1983).

The most familiar examples of such fields are the fields ${\mathbb{Z}}_p$ of integers modulo a prime p, but these are not all. In this chapter we give a complete classification of all finite fields. It turns out that a finite field is uniquely determined up to isomorphism by the number of elements that it contains, that this number must be a power of a prime, and that for every prime ...