Finite field

This article gives a basic definition in the following area: field theory
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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)	$\log _{p}q$, 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