8

Galois Fields

8.1 Finite Fields

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 ...

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