Finite Rings - Finite Field

Finite Field

The theory of finite fields is perhaps the most important aspect of finite ring theory due to its intimate connections with algebraic geometry, Galois theory and number theory. An important, but fairly old aspect of the theory is the classification of finite fields (Jacobson 1985, p. 287):

• The order or number of elements of a finite field equals pn, where p is a prime number called the characteristic of the field, and n is a positive integer.
• For every prime number p and positive integer n, there exists a finite field with pn elements.
• Any two finite fields with the same order are isomorphic.

Despite the classification, finite fields are still an active area of research, including recent results on the Kakeya conjecture and open problems regarding the size of smallest primitive roots (in number theory).