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).

Particular cases

Field size $q$ Underlying prime $p$ (field characteristic) $\log_pq$, i.e, the number $r$ such that $q = p^r$ Field
2 2 1 field:F2
3 3 1 field:F3
4 2 2 field:F4
5 5 1 field:F5
7 7 1 field:F7
8 2 3 field:F8
9 3 2 field:F9
11 11 1 field:F11
13 13 1 field:F13
16 2 4 field:F16
17 17 1 field:F17