The GF command returns a module G of procedures and constants for doing arithmetic in the finite field GF(p^k), a Galois Field with  elements. The field GF(p^k) is defined by the field extension GF(p)[x]/(a) where a is an irreducible polynomial of degree k over the integers mod p.

If a is not specified, an irreducible polynomial of degree k over the integers mod p is chosen at random.  It can be accessed as the constant G:-extension.  The elements of GF(p^k) are represented using the  representation.

First, you need to define an instance of a Galois field using, for example, G := GF(2, 3). This defines all operations for G, the field of characteristic 2 with 8 elements.

The G:-input and G:-output commands convert from an integer in the range  to the corresponding polynomial and back. Alternatively, G:-ConvertIn and G:-ConvertOut convert an element from GF(p^k) to a Maple sum of products, a univariate polynomial where the variable used is that given in the argument a. Otherwise the name `?` is used.

Arithmetic in the field is defined by the following functions. G:-`+`, G:-`-`, G:-`*`, G:-`^`, G:-inverse, G:-`/`

Field arithmetic can be written very naturally by using a use statement; see the examples below.

The additive and multiplicative identities are given by G:-zero and G:-one.

The G:-trace, G:-norm, and G:-order commands compute the trace, norm and multiplicative order of an element from GF(p^k) respectively.

The G:-random command returns a random element from GF(p^k).

The G:-PrimitiveElement command generates a primitive element at random.

The G:-isPrimitiveElement command tests whether an element from GF(p^k) is a primitive element, being a generator for the multiplicative group GF(p^k) - {0}.

For backwards compatibility, exports of the module returned by the GF command can also be accessed by indexed notation, such as G[':-ConvertIn']. However, using the more modern form G:-ConvertIn, you do not need to quote the name to avoid evaluation.