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Dom::GaloisField -- finite fields

Introduction

Dom::GaloisField(p, n, f) creates the residue class field Zp[X]/⟨f⟩, a finite field with pn elements. If f is not given, it is chosen at random among all irreducible polynomials of degree n.

Dom::GaloisField(q) (where q=pn) is equivalent to Dom::GaloisField(p,n).

Dom::GaloisField(F, n, f) creates the residue class field F[X]/⟨f⟩, a finite field with |F| n elements. If f is not given, it is chosen at random among all irreducible polynomials of degree n.

Domain

Dom::GaloisField(q)
Dom::GaloisField(p, n)
Dom::GaloisField(p, n, f)
Dom::GaloisField(F, n)
Dom::GaloisField(F, n, f)

Parameters

q - prime power
p - prime
n - positive integer
f - univariate irreducible polynomial over Dom::IntegerMod(p) or F, or polynomial expression convertible to such
F - finite field of type Dom::IntegerMod or Dom::GaloisField.

Details

Introduction

Dom::GaloisField(p,n,f)(g) (or, respectively, Dom::GaloisField(F,n,f)(g)) creates the residue class of g modulo f. It is represented by the unique polynomial in that class that has smaller degree than f.

Creating Elements

Dom::GaloisField(p, n, f)(g)

Parameters

g - univariate polynomial over the ground field in the same variable as f, or polynomial expression convertible to such

Categories

Cat::Field

, Cat::Algebra(F), Cat::VectorSpace(F)

Related Domains

Dom::AlgebraicExtension, Dom::IntegerMod

Entries

zero

the zero element of the field

one

the unit element of the field

characteristic

the characteristic of the field

size

the number of elements of the field

PrimeField

the prime field, which equals Dom::IntegerMod(p).

Variable

the variable of the polynomial f.

VariablesInUse

a list consisting of "Variable" and the variables used by the ground field.

companionMatrix

an n times n-matrix over the ground field, where n is the degree of the field over its ground field. It can be used for representing field elements as matrices since its minimal polynomial is f.

companionPowers

a list of the first n-1 powers of the companion matrix.

Method iszero: test for zero

Method _power: integer power of an element

Method frobenius: Frobenius map

Method conjugates: conjugates of an element

Method order: order of an element

Method isSquare: test whether an element is a square

Method ln: discrete logarithm

Method elementNumber: enumerate field elements

Method matrixRepresentation: isomorphism to the algebra generated by the companion matrix

Method randomPrimitive: choose a primitive element at random

Method isBasis: tests elements for being a basis over the ground field

Method isNormal: tests whether a given field element is normal

Method randomNormal: choose normal element at random

Method isPrimitivePolynomial: tests whether a polynomial over the field is primitive

Method convert: conversion from other types

Method convert_to: conversion to other types

Example 1

We define L to be the field with 4 elements. Then a4=a for every a ∈L, by a well-known theorem.

>> L:=Dom::GaloisField(2, 2, u^2+u+1): L(u+1)^4
                                   u + 1

Super-Domain

Dom::AlgebraicExtension

Axioms

Ax::canonicalRep

Changes




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