# Structure Operations

## Related Structures

### `Category(R): RngGal -> Cat`

### `Parent(R): RngGal -> PowerStructure`

### `Centre(R): RngGal -> RngGal`

### `PrimeRing(R): RngGal -> RngGal`

### `PrimeField(R): RngGal -> RngGal`

### `FieldOfFractions(R): RngGal -> RngGal`

## 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](../../BasicRings/IntroductionToRings/generic.md#rngintro-predb).

### `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`