Structure Operations

Numerical Invariants

Characteristic(R): RngGal -> RngIntElt: The characteristic of \(R\), which is \(p^a\) where \(R\) is considered as \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\).

# R: RngGal -> RngIntElt: The cardinality of \(R\), which is \(p^{ad}\) where \(R\) is considered as \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\) and \(d\) is the degree of \(D\).

Degree(R): RngGal -> RngIntElt: The degree of \(R\), which is the degree of \(D\), where \(R\) is considered as \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\).

ResidueField(R): RngGal -> FldFin: The residue field of \(R\), which is the finite field \({\mathbb{Z}}_p[x]/\langle D\rangle\), where \(R\) is considered as \({\mathbb{Z}}_{p^a}[x]/\langle D\rangle\).

Ring Predicates and Booleans

The following functions are described for rings in general in Section Predicates and Boolean Operations.

IsCommutative(R): RngGal -> BoolElt

IsUnitary(R): RngGal -> BoolElt

IsFinite(R): RngGal -> BoolElt

IsOrdered(R): RngGal -> BoolElt

IsField(R): RngGal -> BoolElt

IsEuclideanDomain(R): RngGal -> BoolElt

IsPID(R): RngGal -> BoolElt

IsUFD(R): RngGal -> BoolElt

IsDivisionRing(R): RngGal -> BoolElt

IsEuclideanRing(R): RngGal -> BoolElt

IsPrincipalIdealRing(R): RngGal -> BoolElt

IsDomain(R): RngGal -> BoolElt

R eq G: RngGal, Rng -> BoolElt

R ne G: RngGal, Rng -> BoolElt