Operations with Fields

BaseRing(L): RngLocA -> Rng
CoefficientRing(L): RngLocA -> Rng: Return the coefficient field of the local field \(L\). This is the field which was extended to construct \(L\).

DefiningPolynomial(L): RngLocA -> RngUPolElt: Return the polynomial used to define the local field \(L\) as an extension of its coefficient field.

Degree(L): RngLocA -> RngIntElt: Return the degree of the local field \(L\), that is, the degree of its defining polynomial.

Degree(L, R): RngLocA, Rng -> RngIntElt: Return the degree of \(L\) as an extension of \(R\) where \(R\) is some coefficient ring of \(L\).

InertiaDegree(L): RngLocA -> RngIntElt
RamificationDegree(L): RngLocA -> RngIntElt
RamificationIndex(L): RngLocA -> RngIntElt: Return the degree of the inertial or totally ramified subfield of the local field \(L\) as an extension of the coefficient field of \(L\).

Precision(L): RngLocA -> RngIntElt: Return the precision of the local field \(L\). This is the maximum number of digits which can occur in an element of \(L\), the difference between the valuation of an element of \(L\) and the valuation of the term of highest valuation occurring in that element.

Prime(L): RngLocA -> RngElt: Return the prime of the local field \(L\). This is the same as the prime of the coefficient field of \(L\).

Example: Coefficient Ring and Defining Polynomial (ex-ef970a)

Continuing from the first example we have :

> CoefficientRing(L);
7-adic field mod 7^50
> DefiningPolynomial(L);
$.1^6 + (32 + O(7^50))*$.1^5 + (390 + O(7^50))*$.1^4 + (2284 + O(7^50))*$.1^3 +
    (6588 + O(7^50))*$.1^2 + (8744 + O(7^50))*$.1 + 5452 + O(7^50)
> Precision(L);
150
> Prime(L);
7
> Degree(L); RamificationDegree(L); InertiaDegree(L);
6
3
2

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QuotientRepresentation(L): RngLocA -> RngUPolRes: Return the polynomial quotient ring which is isomorphic to the local field \(L\) and is used to represent \(L\).

RamifiedRepresentation(L): RngLocA -> FldPad, Map: Return the local field isomorphic to the local field \(L\) constructed as an unramified then a ramified extension and the map from \(L\) into the isomorphic field.

Example: Two-Step Representation of a Local Field (ex-a41b45)

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We create a \(1\) step local field and compute its representation as a \(2\) step local field.

> P<x> := PolynomialRing(Integers());
> L<a> := LocalField(pAdicField(7, 50), x^6 - 49*x^2 + 686);
> L;
Extension of 7-adic field mod 7^50 by x^6 + O(7^50)*x^5 + O(7^50)*x^4 +
    O(7^50)*x^3 - (7^2 + O(7^52))*x^2 + O(7^50)*x + 2*7^3 + O(7^53)
> QuotientRepresentation(L);
Univariate Quotient Polynomial Algebra in $.1 over 7-adic field mod 7^50
with modulus $.1^6 + O(7^50)*$.1^5 + O(7^50)*$.1^4 + O(7^50)*$.1^3 - (7^2 +
    O(7^52))*$.1^2 + O(7^50)*$.1 + 2*7^3 + O(7^53)
> RR, m := RamifiedRepresentation(L);
> RR;
Totally ramified extension defined by the polynomial x^2 +
    417092732355694537113348201703437033788663*$.1^2 +
    586194602218356762336379252895075698537137*$.1 -
    130094113224633998166887533755901214096597
 over Unramified extension defined by the polynomial x^3 + 6*x + 2
 over 7-adic field mod 7^50
> m(L.1);
RR.1 + O(RR.1^92)
> RR.1 @@ m;
O(7^48)*$.1^5 + O(7^48)*$.1^4 + O(7^49)*$.1^3 + O(7^49)*$.1^2 + $.1 + O(7^50)
> CoefficientRing(RR).1 @@ m;
O(7^34)*$.1^5 - (954564700580430506024960512238*7^-1 + O(7^35))*$.1^4 +
    O(7^36)*$.1^3 + (1031213687115590174398504554631*7^-1 + O(7^35))*$.1^2 +
    O(7^37)*$.1 + 131284877366067295106350173568*7 + O(7^36)

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AssignNames(~L, S): RngLocA, SeqEnum[MonStgElt]: Assign the name in the sequence \(S\) to the generator of the extension defining the local field \(L\).

Name(L, i): RngLocA, RngIntElt -> RngLocAElt: Return the generator of the local field \(L\) which has assigned to it the name in the sequence \(S\) which was input to AssignNames. The only valid input for \(i\) is \(1\).

Discriminant(L): RngLocA -> RngElt: Return the discriminant of the local field \(L\).

ResidueClassField(L): RngLocA -> Rng, Map: Return the residue class field of the maximal order of the local field \(L\) and the map between \(L\) and its residue class field.

RelativeField(L, m): RngLocA, Map -> RngLocA, Map, Map: Return \(L\) as an extension of the domain of the map \(m\) which should be a map from a subfield of \(L\) (having the same coefficient ring as \(L\)) into \(L\).

Predicates on Fields

IsRamified(L): RngLocA -> BoolElt: Return whether the local field \(L\) has a non trivial ramified subfield, that is, the ramification degree of \(L\) is greater than \(1\).

IsTamelyRamified(L): RngLocA -> BoolElt
IsWildlyRamified(L): RngLocA -> BoolElt: Return whether the local field \(L\) is tamely or wildly ramified.

IsTotallyRamified(L): RngLocA -> BoolElt: Return whether the local field \(L\) is a totally ramified extension, that is, \(L\) has a trivial inertial subfield.

IsUnramified(L): RngLocA -> BoolElt: Return whether the local field \(L\) is equal to its inertial subfield.

Operations with Fields#

Predicates on Fields#

Operations with Fields

Predicates on Fields