## Chapter 22 Finite Fields

Finite fields appear in many applications of algebra, including coding theory and cryptography. We already know one finite field, \({\mathbb Z}_p\text{,}\) where \(p\) is prime. In this chapter we will show that a unique finite field of order \(p^n\) exists for every prime \(p\text{,}\) where \(n\) is a positive integer. Finite fields are also called Galois fields in honor of Évariste Galois, who was one of the first mathematicians to investigate them.