finite field
Most fields you meet — the rationals, the reals, the complex numbers — have infinitely many elements. A finite field is a complete number system with only finitely many elements, yet you can still add, subtract, multiply and divide (by anything except zero) exactly as you expect. The simplest examples are clock arithmetics with a prime number of hours.
Precisely, a finite field is a field with a finite number of elements. A remarkable classification holds: the number of elements is always a prime power p^n, one finite field exists for each such power, and any two finite fields of the same size are isomorphic. The field with p^n elements is written GF(p^n) or F(p^n) — the GF honoring Galois.
The smallest finite fields are the integers modulo a prime p, where arithmetic wraps around at p. Larger ones, of size p^n with n > 1, are built as extension fields of these. Finite fields are not just curiosities: they underpin error-correcting codes, cryptography and much of digital communication.
The field with 2 elements, GF(2) = {0, 1}, has 1 + 1 = 0 and is the arithmetic of bits. The field GF(4) has 4 elements and is built as a degree-2 extension of GF(2).
GF(2) is the two-element field of bits.
There is no finite field with, say, 6 or 10 elements, because 6 and 10 are not prime powers. The multiplicative group of any finite field (all its nonzero elements) is cyclic.