Finite Field - Classification

Classification

The finite fields are classified as follows (Jacobson 2009, §4.13,p. 287):

• The order, or number of elements, of a finite field is of the form 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 number of elements are isomorphic. That is, under some renaming of the elements of one of these, both its addition and multiplication tables become identical to the corresponding tables of the other one.

This classification justifies using a naming scheme for finite fields that specifies only the order of the field. One notation for a finite field is or F_pn. Another notation is GF(pn), where the letters "GF" stand for "Galois field".