The theory of finite fields, whose origins can be traced back to the works of [[Gauss]] and [[Galois]], has played a part in various branches of mathematics. Due to the applicability of the concept in other topics of mathematics and sciences like computer science there has been a resurgence of interest in finite fields and this is partly due to important applications in [[coding theory]] and [[cryptography]]. Applications of finite fields introduce some of these developments in [[cryptography]], [[computer algebra]] and [[coding theory]].
finite field or [[Galois field]] is a field with a [[Wikt:finite|finite]] order (number of elements). The order of a finite field is always a [[prime]] or a power of prime. For each [[prime power]] ''q'' = ''p<sup>r</sup>'' , there exists exactly one finite field with ''q'' elements, [[up to]] isomorphism. This field is denoted ''GF''(''q'') or '''F'''<sub>''q''</sub>. If ''p'' is prime, ''GF''(''p'') is the [[ prime field]] of order ''p''; it is the field of [[residue class#Ring of congruence classes| residue class]] es modulo ''p'' , and its ''p'' elements are denoted 0, 1, ..., ''p''−1. Thus ''a'' = ''b'' in ''GF''(''p'') means the same as ''a'' ≡ ''b'' (mod ''p'').
Many algorithms for factoring polynomials over finite fields include the following three stages: