Finite Fields
A finite field (also called a Galois field) 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 Algebr,
Second Edition, New York, Addison-Wesley Publishing Company, Inc. 1984,
ISBN 0 201 05476 6; or R. Lidl, H. Niederreiter, Finite Fields,
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 prime field,
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.
