We have applied the notion of a finite field as a finite vector space in previous chapters. As a structure, a field has two operations, denoted by + and ∗, which are not necessarily ordinary addition and multiplication. Under the operation + all elements of a field form a commutative group whose identity is denoted by 0 and inverse of a by −a. Under the operation ∗, all elements of the field form another commutative group with identity denoted by 1 and the inverse of a by a−1. Note that the element 0 has no inverse under ∗. There is also a distributive identity that links + and ∗, such that a ∗ (b+ c) = (a ∗ b)+ (a ∗ c) for all field elements a, b and c. This identity is the same as the cancellation property: If c 6= 0 and a∗c = b∗c, then a = b. The following groups represent fields:

1. The set Z of rational numbers, or the set R of real numbers, or the set C of complex numbers, are all infinite fields under ordinary addition and multiplication.