Finite fields and their applications

Abstract

This chapter discusses finite fields and their applications. It also traces the historical development of finite fields, outlining some of their basic properties. In general, F q denotes the finite field of order q . The general theory of finite fields may be said to begin with the work of Carl Friedrich Gauss and Evariste Galois. Galois supposes Fx to be irreducible mod p and of degree v and solves Fx ≡ 0 by introducing new symbols, which might be just as useful as the imaginary unit i in analysis. Galois approach via imaginary roots and Dedekind's approach via residue class rings are shown to be essentially equivalent by Kronecker. A careful choice of the representation of a finite field F q may assist in the algorithms for the implementation of arithmetic operations in F q . Enumeration theorems for ordered bases of various types are known. Irreducible polynomials of degree n over F q are important for the construction of the field F q n .