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#