Applications
Discrete exponentiation, also known as calculating a = xn from x and n, can be computed quickly using techniques of fast exponentiation such as binary exponentiation, which takes only O(log n) field operations. No fast way of computing the discrete logarithm n given a and x is known, and this has many applications in cryptography, such as the Diffie-Hellman protocol.
Finite fields also find applications in coding theory: many codes are constructed as subspaces of vector spaces over finite fields.
Within number theory, the significance of finite fields is their role in the definition of the Frobenius element (or, more accurately, Frobenius conjugacy class) attached to a prime ideal in a Galois extension of number fields, which in turn is needed to make sense of Artin L-functions of representations of the Galois group, the non-abelian generalization of Dirichlet L-functions.
Counting solutions to equations over finite fields leads into deep questions in algebraic geometry, the Weil conjectures, and in fact was the motivation for Grothendieck's development of modern algebraic geometry.
Read more about this topic: Finite Field