Finite Fields

A <sl>finite field</sl> (also called a <sl>Galois field</sl>) is a finite algebraic structure where on can add, multiply, and divide under the same laws (for example, commutativity, associativity, or distributivity) as apply to the rational, real, or complex numbers. Unlike those three fields, for any finite field there exists a positive prime integer p, called the characteristic, such that p*x=0 for any element x in the finite field. In fact, the number of elements in a finite filed is a power of the characteristic and for each prime p and positive integer n there exists exactly one finite field with p**n elements, up to an isomorphism. (For more information about the algebraic structure and properties of finite fields, see for example, S. Lang <sl>Algebr</sl>, Second Edition, New York, Addison-Wesley Publishing Company, Inc. 1984, ISBN 0 201 05476 6; or R. Lidl, H. Niederreiter, <sl>Finite Fields</sl>, Encyclopedia of Mathematics and Its Applications, Vol. 20, Cambridge. Cambridge Univ. Press, 1983, ISBN 0 521 30240 4) When n=1, the field has p elements and is called a <sl>prime field</sl>, discussed in Modular Arithmetic and Prime Fields in section 8.11.1. There are several ways of implementing extensions of finite fields, and Axiom provides quite a bit of freedom to allow you to choose the one that is best for your application. Moreover, we provide operations for converting among the different representations of extensions and different extensions of a single field. Finally, note that you usually need to package call operations from finite fields if the operations do not take as an argument an object of the field. See Package Calling and Target Types in section 2.9 for more information on package calling.

- Modular Arithmetic and Prime Fields
- Extensions of Finite Fields
- Irreducible Modulus Polynomial Representations
- Cyclic Group Representations
- Normal Basis Representations
- Conversion Operations for Finite Fields
- Utility Operations for Finite Fields