Finite field
From Groupprops
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 , the unique finite field of size
by the symbols
or
. (Note that GF stands for Galois field in recognition of Galois's pioneering work in field theory).
Particular cases
Field size ![]() |
Underlying prime ![]() |
![]() ![]() ![]() |
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 |