ABSTRACT
Fields with 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. The most familiar examples of such fields are the 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 power there exists a field with that number of elements. All these facts were discovered by Galois.
