Abstract Algebra: Theory and Applications

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.