In abstract algebra, a **finite field** or **Galois field** (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory. The finite fields are completely known, as will be described below. ## The complete list
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*), <*f*(*T*)> denotes the ring ideal generated by *f*(*T*), 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.
## Examples The polynomial *f*(*T*) = *T*^{ 2} + *T* + 1 is irreducible over **GF**(2), and **GF**(4)=**GF**(2)/<*T*^{2}+*T*+1> can therefore be written as the set {0, 1, *t*, *t*+1} where the multiplication is defined (modularly) by *t*^{2} + *t* + 1 = 0. For example, to determine *t*^{3}, note that *t*(*t*^{2} + *t* + 1) = 0; so *t*^{3} + *t*^{2} + *t* = 0, and thus *t*^{3} + *t*^{2} + *t* + 1 = 1, so *t*^{3} = 1. Similarly, since the characteristic of the field is 2 - coefficients are in **GF**(2), we can calculate powers of *t* in this instance can be calculated by noting first that *t*^{2}+*t*+1=0, and then *t*^{2}=*t*+1. Then *t*^{3} = *t*(*t*^{2}) = *t*(*t*+1) = *t*^{2}+*t* = (*t*+1)+*t* = 1 as before. In order to find the multiplicative inverse of *t* in this field, we have to find a polynomial *p*(*T*) such that *T* * *p*(*T*) = 1 modulo *T*^{ 2} + *T* + 1. The polynomial *p*(*T*) = *T* + 1 works, and hence 1/*t* = *t* + 1. Note that the field **GF**(4) is completely unrelated to the ring **Z**_{4} of integers modulo 4. 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.
## Properties and facts 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 homomorphism *f* : *F* → *F* defined by *f*(*x*) = *x*^{p} is bijective, and is therefore an automorphism. It is called the Frobenius automorphism, after Ferdinand Georg Frobenius. The Frobenius automorphism has order *n*, so that the cyclic group it generates is the full group of automorphisms of the field. The field **GF**(*p*^{m}) contains a copy of **GF**(*p*^{n}) if and only if *n* divides *m*. The reason for this is that there exist irreducible polynomials of every degree over **GF**(*p*^{n}). If we actually construct our finite fields in such a fashion that **GF**(*p*^{n}) *is contained in* **GF**(*p*^{m}) whenever *n* divides *m*, then we can form the union of all these fields. This union is also a field, albeit an infinite one. It is the algebraic closure of each of the fields **GF**(*p*^{n}).
## Applications 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 subspaces of vector spaces over finite fields.
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