Table of contents |

2 Examples 3 Properties and facts 4 Applications |

Since every field of characteristic 0 contains the rationals and is therefore infinite, all finite fields have prime characteristic.

If *p* is a prime, the integers modulo *p* form a field with *p* elements, denoted by **Z**_{p}, **F**_{p} or **GF**(*p*). Every other field with *p* elements is isomorphic to this one.

If *q* = *p*^{n} is a prime power, then there exists up to isomorphism exactly one field with *q* elements, written as **F**_{q} or **GF**(*q*). It can be constructed as follows: find an irreducible polynomial *f*(*T*) of degree *n* with coefficients in **GF**(*p*), then define **GF**(*q*) = **GF**(*p*)[*T*] / <*f*(*T*)>. Here, **GF**(*p*)[*T*] denotes the ring of all polynomials with coefficients in **GF**(*p*), and the quotient is meant in the sense of factor rings - the set of polynomials with coefficients in **GF**(*p*) on division by *f*(*T*). The polynomial *f*(*T*) can be found by factoring the polynomial *T*^{ q}-*T* over **GF**(*p*). The field **GF**(*q*) contains **GF**(*p*) as a subfield.

There are no other finite fields.

To construct the field **GF**(27), we start with the irreducible polynomial *T*^{ 3} + *T*^{ 2} + *T* - 1 over **GF**(3). We then have **GF**(27) = {*at*^{2} + *bt* + *c* : *a*, *b*, *c* in **GF**(3)}, where the multiplication is defined by *t*^{ 3} + *t*^{ 2} + *t* - 1 = 0, or working from the rearrangement of the above in isolating the *t*^{3} term.

If *F* is a finite field with *q* = *p*^{n} elements (where *p* is prime), then *x*^{q} = *x* for all *x* in *F*.
Furthermore, the Frobenius homomorphism *f* : *F* `->` *F* defined by *f*(*x*) = *x*^{p} is bijective, and is therefore an automorphism.
The Frobenius homomorphism has order *n*, and the cyclic group it generates is the full group of automorphisms of the field.

The field **GF**(*p ^{m}*) contains a copy of

If we actually construct our finite fields in such a fashion that **GF**(*p ^{n}*)

The multiplicative group of every finite field is cyclic, a special case of a theorem mentioned in the article about fields. This means that if *F* is a finite field with *q* elements, then there always exists an element *x* in *F* such that *F* = { 0, 1, *x*, *x*^{2}, ..., *x*^{q-2} }.

The element *x* is not unique. If we fix one, then for any non-zero element *a* in *F*_{q}, there is a unique integer *n* in {0, ..., *q* - 2} such that *a* = *x*^{n}. The value of *n* for a given *a* is called the * discrete log* of *a* (in the given field, to base *x*). In practice, although calculating *x*^{n} is relatively trivial given *n*, finding *n* for a given *a* is (under current theories) a computationally difficult process, and so has many applications in cryptography.

Finite fields also find applications in coding theory: many codes are constructed as subspacess of vector spaces over finite fields.\n