Finite field

Definition

A finite field is a field with only finitely many elements.

Some key facts about finite fields include:

The characteristic of a finite field must be a prime number.
The size of a finite field must be a prime power. In fact, it must be a power of the prime number that is the characteristic of the field.
For any prime power, there is a unique (up to isomorphism) finite field whose size equals that prime power.

Combining the above key facts, we denote, for any prime power $q$ , the unique finite field of size $q$ by the symbols $\mathbb{F}_q$ or $GF(q)$ . (Note that GF stands for Galois field in recognition of Galois's pioneering work in field theory).