Rings, Fields, and Algebras

Magma supports the following categories of rings, fields and algebras:

RngInt

integer ring

RngIntRes

residue rings of integers (i.e. modulo n)

RngUPol

univariate polynomial rings

RngUPolRes

residue rings of univariate polynomials

RngMPol

distributed multivariate polynomial rings

RngInvar

invariant rings of finite groups

RngPow

power, Laurent and Puiseux series rings

RngLaz

lazy series rings

RngVal

valuation rings

RngPad

p-adic rings and fields

AlgChtr

algebras of characters of finite groups

AlgFP

finitely presented algebras

FldRat

rational field

FldPr, FldRe, FldCom

real and complex fields

FldFin

finite (Galois) fields

FldFunRat

rational function fields

FldFun

function fields

FldNum

number fields

FldQuad

quadratic fields

FldCyc

cyclotomic fields

FldAb

abelian extensions (of number fields)

AlgMat

matrix algebras

Structure Constant Algebras

Associative Algebras

Basic Algebras

Lie Algebras

Group Algebras

They are created by the functions:

• IntegerRing or Integers
• ResidueClassRing
• PolynomialRing or PolynomialAlgebra
• InvariantRing
• PowerSeriesRing or PowerSeriesAlgebra, LaurentSeriesRing or LaurentSeriesAlgebra
• ValuationRing
• pAdicRing, pAdicField
• CharacterRing
• FreeAlgebra
• RationalField
• RealField, ComplexField
• FiniteField or GaloisField or GF
• FieldOfFractions
• NumberField
• QuadraticField
• CyclotomicField
• Local Ring
• MatrixRing or MatrixAlgebra
• Algebra
• AssociativeAlgebra
• BasicAlgebra
• LieAlgebra
• GroupAlgebra

One may create any ring by creating one of these magmas and then taking the required subring or quotient ring, as appropriate, with the sub or quo constructor.

The characteristic of ring R is Characteristic(R). Its cardinality is #R.

The operators for arithmetic on ring elements are: r + s, r - s, -r, r * s, and r ^ n. In fields the division operator is / and in other rings division-with-remainder is performed with div and mod .

The additive identity is Zero(R) or R!0, and the multiplicative identity is One(R) or R!1 .

Functions testing properties of a ring element include IsZero, IsOne, IsMinusOne, IsUnit.

For rings in which an ordering is defined on the elements, the comparison operations are gt, ge, lt, le, and also Max and Min.

Example

> GF81<w> := GF(3, 4); 
> print GF81;
Finite field of size 3^4
> e := 2 * w^5; print e;
w^45

> P<x> := PolynomialRing(GF81);                                                
> print P;
Univariate Polynomial Ring in x over GF(3^4)
> p := 2*x^6 - w*x^2 + 1; print p;
2*x^6 + w^41*x^2 + 1

> FF := FieldOfFractions(P); print FF;
Rational function field of rank 1 over GF(3^4)
Variables: $.1
> print (p^2 - FF!1) / (p + 1) eq (p - 1);
true