Elementary Invariants

These functions return some simple information partially defining a local ring.

Prime(L): RngPad -> RngIntElt
Prime(L): FldPad -> RngIntElt
Prime(L): RngPadRes -> RngIntElt
Prime(L): RngPadResExt -> RngIntElt: Given a local ring or field \(L\), return the prime \(p\) defining the \(p\)-adic ring or field underlying \(L\). This is also the characteristic of the residue class field of \(L\).

InertiaDegree(L): RngPad -> RngIntElt
InertiaDegree(L): FldPad -> RngIntElt
InertiaDegree(L): RngPadRes -> RngIntElt
InertiaDegree(L): RngPadResExt -> RngIntElt: Return the inertia degree of the local ring or field \(L\) over its coefficient ring.

InertiaDegree(K, L): RngPad, RngPad -> RngIntElt
InertiaDegree(K, L): FldPad, FldPad -> RngIntElt
InertiaDegree(K, L): RngPadRes, RngPadRes -> RngIntElt
InertiaDegree(K, L): RngPadResExt, RngPadResExt -> RngIntElt: Return the inertia degree of the local ring or field \(K\) relative to its subring \(L\).

AbsoluteInertiaDegree(L): RngPad -> RngIntElt
AbsoluteInertiaDegree(L): FldPad -> RngIntElt
AbsoluteInertiaDegree(L): RngPadRes -> RngIntElt
AbsoluteInertiaDegree(L): RngPadResExt -> RngIntElt
AbsoluteInertiaIndex(L): RngPad -> RngIntElt
AbsoluteInertiaIndex(L): FldPad -> RngIntElt
AbsoluteInertiaIndex(L): RngPadRes -> RngIntElt
AbsoluteInertiaIndex(L): RngPadResExt -> RngIntElt: Return the inertia degree of the local ring or field \(L\) over the \(p\)-adic ring.

RamificationDegree(L): RngPad -> RngIntElt
RamificationDegree(L): FldPad -> RngIntElt
RamificationDegree(L): RngPadRes -> RngIntElt
RamificationDegree(L): RngPadResExt -> RngIntElt
RamificationIndex(L): RngPad -> RngIntElt
RamificationIndex(L): FldPad -> RngIntElt
RamificationIndex(L): RngPadRes -> RngIntElt
RamificationIndex(L): RngPadResExt -> RngIntElt: Return the ramification degree of the local ring or field \(L\) over its coefficient ring.

RamificationDegree(K, L): RngPad, RngPad -> RngIntElt
RamificationDegree(K, L): FldPad, FldPad -> RngIntElt
RamificationDegree(K, L): RngPadRes, RngPadRes -> RngIntElt
RamificationDegree(K, L): RngPadResExt, RngPadResExt -> RngIntElt
RamificationIndex(K, L): RngPad, RngPad -> RngIntElt
RamificationIndex(K, L): FldPad, FldPad -> RngIntElt
RamificationIndex(K, L): RngPadRes, RngPadRes -> RngIntElt
RamificationIndex(K, L): RngPadResExt, RngPadResExt -> RngIntElt: Return the ramification degree of the local ring or field \(K\) relative to its subring \(L\).

AbsoluteRamificationDegree(L): RngPad -> RngIntElt
AbsoluteRamificationDegree(L): FldPad -> RngIntElt
AbsoluteRamificationDegree(L): RngPadRes -> RngIntElt
AbsoluteRamificationDegree(L): RngPadResExt -> RngIntElt
AbsoluteRamificationIndex(L): RngPad -> RngIntElt
AbsoluteRamificationIndex(L): FldPad -> RngIntElt
AbsoluteRamificationIndex(L): RngPadRes -> RngIntElt
AbsoluteRamificationIndex(L): RngPadResExt -> RngIntElt: Return the ramification degree of the local ring or field \(L\) over the \(p\)-adic ring.

AbsoluteDegree(L): RngPad -> RngIntElt: The degree of \(L\) over \({\mathbb{Z}}_p\).

Degree(L): RngPad -> RngIntElt
Degree(L): FldPad -> RngIntElt
Degree(L): RngPadRes -> RngIntElt
Degree(L): RngPadResExt -> RngIntElt: Return the degree of the local ring or field \(L\) over its coefficient ring.

Degree(K, L): RngPad, RngPad -> RngIntElt
Degree(K, L): FldPad, FldPad -> RngIntElt
Degree(K, L): RngPadRes, RngPadRes -> RngIntElt
Degree(K, L): RngPadResExt, RngPadResExt -> RngIntElt: Return the degree of the local ring or field \(K\) relative to its subring \(L\).

DefiningPolynomial(L): RngPad -> RngUPolElt
DefiningPolynomial(L): FldPad -> RngUPolElt
DefiningPolynomial(L): RngPadRes -> RngUPolElt
DefiningPolynomial(L): RngPadResExt -> RngUPolElt: Return the minimal polynomial of the generator of \(L\) over its coefficient ring. If \(L\) is \(p\)-adic, the polynomial \(x - 1\) is returned. For free precision rings and fields, the coefficients of the defining polynomial are given to the default precision of \(L\).

DefiningPolynomial(K, L): RngPad, RngPad -> RngUPolElt
DefiningPolynomial(K, L): FldPad, FldPad -> RngUPolElt
DefiningPolynomial(K, L): RngPadRes, RngPadRes -> RngUPolElt
DefiningPolynomial(K, L): RngPadResExt, RngPadRes -> RngUPolElt
DefiningPolynomial(K, L): RngPadResExt, RngPadResExt -> RngUPolElt: Return the minimal polynomial of the generator of \(K\) over its coefficient ring \(L\).

DefiningMap(L): RngPad -> Map
DefiningMap(L): FldPad -> Map: Given a free precision local ring or field \(L\), return the map that was used to define the extension (see Section Creation of Unbounded Precision Extensions for information on defining extension by maps). If a map was not used, then an error is raised.

HasDefiningMap(L): RngPad -> BoolElt, Map
DefiningMap(L): FldPad -> BoolElt, Map: Given a free precision local ring or field \(L\), return true if \(L\) is defined by a map; if so, the defining map is also returned.

PrimeRing(L): RngPad -> RngPad
PrimeRing(L): RngPadRes -> RngPadRes
PrimeRing(L): RngPadResExt -> RngPadRes
PrimeField(L): FldPad -> FldPad
pAdicRing(L): RngPad -> RngPad
pAdicRing(L): RngPadRes -> RngPadRes
pAdicRing(L): RngPadResExt -> RngPadRes
pAdicField(L): FldPad -> FldPad: Given a local ring or field \(L\), return the \(p\)-adic ring or field which is a subring of \(L\).

BaseRing(L): RngPad -> RngPad
BaseRing(L): RngPadRes -> RngPadRes
BaseRing(L): RngPadResExt -> Rng
CoefficientRing(L): RngPad -> RngPad
CoefficientRing(L): RngPadRes -> RngPadRes
CoefficientRing(L): RngPadResExt -> Rng
BaseField(L): FldPad -> FldPad
CoefficientField(L): FldPad -> FldPad
BaseRing(L): FldPad -> FldPad: Given a local ring or field \(L\), return the base ring of \(L\).

ResidueClassField(L): RngPad -> FldFin, Map
ResidueClassField(L): RngPadRes -> FldFin, Map
ResidueClassField(L): RngPadResExt -> FldFin, Map: Given a local ring or field \(L\), return the residue class field \(K\) of \(L\), and a map from \(L\) to \(K\).

ResidueSystem(R): RngPad -> [RngPadElt]
ResidueSystem(R): RngPadRes -> [RngPadEltRes]
ResidueSystem(R): RngPadResExt -> [RngPadEltResExt]
ResidueSystem(R): FldPad -> [FldPadElt]: Given a \(p\)-adic ring or field \(R\), compute a set of representatives of the residue class field of \(R\) as elements of \(R\).

UniformizingElement(L): RngPad -> RngPadElt
UniformizingElement(L): RngPadRes -> RngPadResElt
UniformizingElement(L): RngPadResExt -> RngPadResExtElt
UniformizingElement(L): FldPad -> FldPadElt: Given a local ring or field \(L\), return the uniformizing element of \(L\).

L . 1: RngPad -> RngPadElt
L . 1: RngPadRes -> RngPadResElt
L . 1: RngPadResExt -> RngPadResExtElt
L . 1: FldPad -> FldPadElt: Given a local ring or field \(L\), return an element \(\alpha\) of \(L\) such that if \(K\) is \(L\)’s base ring or field, then the powers of \(\alpha\) give a basis of \(L\) as a vector space over \(K\).

Precision(L): RngPad -> RngIntElt
Precision(L): RngPadRes -> RngIntElt
Precision(L): RngPadResExt -> RngIntElt
Precision(L): FldPad -> RngIntElt: Given a local ring or field \(L\), return the precision with which \(L\) has been created. If \(L\) is a local ring this is the maximum absolute precision to which its elements can be created. If \(L\) is a local field this is the maximum relative precision to which its elements can be created. If \(L\) is an unbounded free precision ring or field, then infinity is returned.

HasPRoot(R): RngPad -> BoolElt: Given a local ring \(R\) extending \({\mathbb{Z}}_p\) for some prime \(p\), decide if \(R\) contains a primitive \(p\)-th root of unity.

HasRootOfUnity(L, n): RngPad, RngIntElt -> BoolElt: Given a local ring \(L\) and some positive integer \(n\), decide if \(L\) contains a primitive \(n\)th root of unity.

Discriminant(R): RngPad -> RngPadElt: Compute the discriminant of the local ring \(R\) over its coefficient ring. Since \(R\) is defined by either an inertial polynomial or an Eisenstein one, this is equivalent to computing the discriminant of the defining polynomial.

Discriminant(K, k): RngPad, RngPad -> RngPadElt: Given \(p\)-adic rings \(K/k\), compute the discriminant of \(K\) as an extension of \(k\).

AdditiveGroup(R): RngPadRes -> GrpAb, Map: The additive group of the \(p\)-adic quotient \(R\) as an abelian group and the isomorphism from this group back to \(R\).

Example: Elementary Invariants of Local Rings (ex-899921)

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We illustrate the functions in this section for rings. Similar constructions can be used for fields.

> Zp := pAdicRing(5, 20);
> I<a> := UnramifiedExtension(Zp, 3);
> R<x> := PolynomialRing(I);
> L<b> := ext<I | x^3 + 5*a*x^2 + 5>;
> Prime(L);
5
> InertiaDegree(L);
1

The inertia degree of \(L\) is returned as 1 because \(L\) has been defined as a totally ramified extension of \(I\). However, the inertia degree of \(L\) over \(Z_p\) is 3, because \(I\) itself is an unramified extension of \(Z_p\).

> InertiaDegree(L, Zp);
3
> Degree(L);
3
> Degree(L, Zp);
9
> DefiningPolynomial(L);
x^3 + 5*a*x^2 + 5
> P<y> := PolynomialRing(Zp);
> DefiningPolynomial(I);
y^3 + 3*y + 3
> BaseRing(L);
Unramified extension defined by the polynomial x^3 + 3*x + 3
 over 5-adic ring mod 5^20
> PrimeRing(L);
5-adic ring mod 5^20
> PrimeRing(I);
5-adic ring mod 5^20
> ResidueClassField(L);
Finite field of size 5^3
Mapping from: RngPad: L to GF(5^3)
> ResidueClassField(I);
Finite field of size 5^3
Mapping from: RngPad: I to GF(5^3)

Here, we see that the residue class fields of \(I\) and \(L\) are identical. This is due to the fact that \(L\) is a totally ramified extension of \(I\).

> UniformizingElement(L);
b
> Precision(L);
60
> Precision(I);
20
> R<a> := ext<pAdicRing(2) | 2>;
> DefiningPolynomial(R);
(1 + O(2^20))*$.1^2 + (1 + O(2^20))*$.1 + 1 + O(2^20)
> Precision(R);
Infinity

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AbsoluteRootNumber(K): FldPad -> FldCycElt: The root number of a \(p\)-adic extension \(K/{{\mathbb{Q}}}_p\) can be computed as in §3.3 of [JR06], with the result returned (using their convention and not Deligne’s) as an element of \({{\mathbb{Q}}}(\zeta_4)\).

RootNumber(K): FldPad -> FldCycElt: The root number of a relative \(p\)-adic extension can be computed by using AbsoluteRootNumber and an induction relation.

Example: Root Numbers of p-adic Extensions (ex-f6c8be)

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We compute the root numbers of some \(p\)-adic extensions.

> K := ext<pAdicField(2,20)|Polynomial([10,0,1])>; K;
Totally ramified extension defined by the polynomial x^2 + 10
 over 2-adic field mod 2^20
> RootNumber(K);
-zeta_4
> K := pAdicField(3,40);
> A := AllExtensions(K,3);
> _<x> := PolynomialRing(Integers(K)); // printing
> [<RootNumber(FieldOfFractions(a)),DefiningPolynomial(a)> : a in A];
[
    <zeta_4, x^3 + 3*x + 3>,
    <-zeta_4, x^3 + 6*x + 3>,
    <1, x^3 + 3*x^2 + 3>,
    <1, x^3 + 6*x^2 + 3>,
    <1, x^3 + 6*x^2 + 12>,
    <1, x^3 + 6*x^2 + 21>,
    <zeta_4, x^3 + 3>,
    <zeta_4, x^3 + 12>,
    <zeta_4, x^3 + 21>,
    <1, x^3 + 2*x + 1>
]

We take an example where the relative and absolute root numbers differ.

> K := pAdicField(3,20);
> L := ext<K|Polynomial([3,0,1])>; // both ramified
> M := ext<L|Polynomial([L.1,0,1])>; // both ramified
> RootNumber(M); // relative for M/L
zeta_4
> AbsoluteRootNumber(M); // absolute for M/Q3
-zeta_4
> RootNumber(L); // intermediate field
zeta_4
> X := AbsoluteTotallyRamifiedExtension(Integers(M));
> RootNumber(FieldOfFractions(X));
-zeta_4

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Elementary Invariants#

Elementary Invariants