Structure Operations

The following generic ring functions are applicable to the ring of integers and its elements.

Related Structures

Category(Z): RngInt -> Cat

Parent(Z): RngInt -> PowerStructure

PrimeRing(Z): RngInt -> RngInt

Center(Z): RngInt -> RngInt

AdditiveGroup(Z): RngInt -> GrpAb, Map

Create the abelian group of integers under addition. This returns an infinite (additive) abelian group \(A\) of rank 1 together with a map from \(A\) to the ring of integers \(Z\), sending \(A.1\) to \(1\).

MultiplicativeGroup(Z): RngInt -> GrpAb, Map

UnitGroup(Z): RngInt -> GrpAb, Map

Create the abelian group of invertible integers, that is, an abelian group isomorphic to the multiplicative subgroup \(\langle -1\rangle\). This returns an (additive) abelian group \(A\) of order \(2\) together with a map from \(A\) to the ring of integers \(Z\), sending \(A.1\) to \(-1\).

ClassGroup(Z): RngInt -> GrpAb, Map

The class group of the ring of \({\mathbb{Z}}\) (which is trivial).

FieldOfFractions(Z): RngInt -> FldRat

Create the field of fractions \({\mathbb{Q}}\) of the ring of rational integers.

sub< Z | n >: RngInt, RngIntElt -> RngInt

Given \(Z\), the ring of integers or an ideal of it, and an element \(n\) of \(Z\), create the ideal \(aZ\cap Z\) of the ring of integers. Note that this creates an ideal, not just a subring.

Numerical Invariants

Characteristic(Z): RngInt -> RngIntElt

Signature(Z): RngInt -> RngIntElt, RngIntElt

The signature of \({\mathbb{Z}}\) as an order of \({\mathbb{Q}}\), i.e. 1, 0.

Ring Predicates and Booleans

IsCommutative(Z): RngInt -> BoolElt

IsUnitary(Z): RngInt -> BoolElt

IsFinite(Z): RngInt -> BoolElt

IsOrdered(Z): RngInt -> BoolElt

IsField(Z): RngInt -> BoolElt

IsEuclideanDomain(Z): RngInt -> BoolElt

IsPID(Z): RngInt -> BoolElt

IsUFD(Z): RngInt -> BoolElt

IsDivisionRing(Z): RngInt -> BoolElt

IsEuclideanRing(Z): RngInt -> BoolElt

IsPrincipalIdealRing(Z): RngInt -> BoolElt

IsDomain(Z): RngInt -> BoolElt

Z eq R: RngInt, Rng -> BoolElt

Z ne R: RngInt, Rng -> BoolElt