A finite field is a field with a finite field order (i.e., number of elements), also called a Galois field. The order of a
finite field is always a prime or a power
of a prime (Birkhoff and Mac Lane 1996). For each
prime power, there exists exactly
one (with the usual caveat that "exactly one" means "exactly one
up to an isomorphism") finite field GF(
), often written as 
in current usage.
GF(
) is called the prime
field of order 
,
and is the field of residue
classes modulo 
,
where the 
elements are denoted 0, 1, ..., 
. 
in GF(
)
means the same as 
.
Note, however, that 
in the ring of residues modulo 4, so 2 has no reciprocal,
and the ring of residues modulo 4 is distinct from the finite
field with four elements. Finite fields are therefore denoted GF(
), instead of GF(
), where 
, for clarity.
The finite field GF(2) consists of elements 0 and 1 which satisfy the following addition and multiplication tables.
 | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 1 | 0 |
 | 0 | 1 |
| 0 | 0 | 0 |
| 1 | 0 | 1 |
If a subset 
of the elements of a finite field 
satisfies the axioms above with the same operators of 
, then 
is called a subfield. Finite
fields are used extensively in the study of error-correcting
codes.
When 
,
GF(
) can be represented
as the field of equivalence
classes of polynomials whose coefficients
belong to GF(
).
Any irreducible polynomial of degree 
yields the same field
up to an isomorphism. For example, for GF(
), the modulus can be taken as 
or 
. Using the modulus 
, the elements of GF(
)--written 0, 
, 
,
...--can be represented as polynomials
with degree less than 3. For instance,
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
Now consider the following table which contains several different representations of the elements of a finite field. The columns are the power, polynomial representation, triples of polynomial representation coefficients (the vector representation), and the binary integer corresponding to the vector representation (the regular representation).
| power | polynomial | vector | regular |
| 0 | 0 | (000) | 0 |
 | 1 | (001) | 1 |
 |  | (010) | 2 |
 |  | (100) | 4 |
 |  | (011) | 3 |
 |  | (110) | 6 |
 |  | (111) | 7 |
 |  | (101) | 5 |
The set of polynomials in the second column is closed under addition and multiplication modulo 
, and these operations on the set satisfy the axioms
of finite field. This particular finite field is said to be an extension field of
degree 3 of GF(2), written GF(
), and the field GF(2) is called the base field of GF(
). If an irreducible
polynomial generates all elements in this way, it is called a primitive
polynomial. For any prime or prime power 
and any positive integer 
, there exists a primitive irreducible
polynomial of degree 
over GF(
).
For any element 
of GF(
),
, and for any nonzero
element 
of GF(
),
. There is a smallest positive
integer 
satisfying the sum condition 
for some element 
in GF(
). This number is called the field
characteristic of the finite field GF(
). The field characteristic
is a prime number for every finite field, and it
is true that
 |
(6)
|
over a finite field with characteristic 
.
