- Introduction
- Background
- Overview of the p-adics in Magma
- Creation of Local Rings and Fields
- Creation Functions for the p-adics
- Creation of Unramified Extensions
- UnramifiedExtension(L, n) : RngPad, RngIntElt -> RngPad
- UnramifiedQuotientRing(K, k) : FldFin, RngIntElt -> Rng
- UnramifiedExtension(L, f) : RngPad, RngUPolElt -> RngPad
- IsInertial(f) : RngUPolElt -> BoolElt
- HasGNB(R, n, t) : RngPad, RngIntElt, RngIntElt -> BoolElt
- CyclotomicUnramifiedExtension(R, f) : FldPad, RngIntElt -> FldPad
- Example RngLoc_el_creation_unram (H48E2)
- Creation of Totally Ramified Extensions
- Creation of Unbounded Precision Extensions
- Creation of Related Rings
- Other Elementary Constructions
- Attributes of Local Rings and Fields
- Elementary Invariants
- Prime(L) : RngPad -> RngIntElt
- InertiaDegree(L) : RngPad -> RngIntElt
- InertiaDegree(K, L) : RngPad, RngPad -> RngIntElt
- AbsoluteInertiaDegree(L) : RngPad -> RngIntElt
- RamificationDegree(L) : RngPad -> RngIntElt
- RamificationDegree(K, L) : RngPad, RngPad -> RngIntElt
- AbsoluteRamificationDegree(L) : RngPad -> RngIntElt
- AbsoluteDegree(L) : RngPad -> RngIntElt
- Degree(L) : RngPad -> RngIntElt
- Degree(K, L) : RngPad, RngPad -> RngIntElt
- DefiningPolynomial(L) : RngPad -> RngUPolElt
- DefiningPolynomial(K, L) : RngPad, RngPad -> RngUPolElt
- DefiningMap(L) : RngPad -> Map
- HasDefiningMap(L) : RngPad -> BoolElt, Map
- PrimeRing(L) : RngPad -> RngPad
- BaseRing(L) : RngPad -> RngPad
- ResidueClassField(L) : RngPad -> FldFin, Map
- ResidueSystem(R) : RngPad -> [RngPadElt]
- UniformizingElement(L) : RngPad -> RngPadElt
- L . 1 : RngPad -> RngPadElt
- Precision(L) : RngPad -> RngIntElt
- HasPRoot(R) : RngPad -> BoolElt
- HasRootOfUnity(L, n) : RngPad, RngIntElt -> BoolElt
- Discriminant(R) : RngPad -> RngPadElt
- Discriminant(K, k) : RngPad, RngPad -> RngPadElt
- AdditiveGroup(R) : RngPadRes -> GrpAb, Map
- Example RngLoc_elinvar (H48E6)
- AbsoluteRootNumber(K) : FldPad -> FldCycElt
- RootNumber(K) : FldPad -> FldCycElt
- Example RngLoc_padic-rootno-ex (H48E7)
- Operations on Structures
- Element Constructions and Conversions
- Operations on Elements
- Arithmetic
- - x : RngPadElt -> RngPadElt
- x + y : RngPadElt, RngPadElt -> RngPadElt
- x - y : RngPadElt, RngPadElt -> RngPadElt
- x * y : RngPadElt, RngPadElt -> RngPadElt
- x ^ k : RngPadElt, RngIntElt -> RngPadElt
- x div y : RngPadElt, RngPadElt -> RngPadElt
- x div:= y : RngPadElt, RngPadElt -> RngPadElt
- x / y : RngPadElt, RngPadElt -> RngPadElt
- IsExactlyDivisible(x, y) : RngPadElt, RngPadElt -> BoolElt, RngPadElt
- Example RngLoc_Division (H48E12)
- Equality and Membership
- Properties
- Precision and Valuation
- Logarithms and Exponentials
- Norm and Trace
- Norm(x) : RngPadElt -> RngPadElt
- Norm(x, R) : RngPadElt, RngPad -> RngPadElt
- Trace(x) : RngPadElt -> RngPadElt
- Trace(x, R) : RngPadElt, RngPad -> RngPadElt
- MinimalPolynomial(x) : RngPadElt -> RngUPolElt
- MinimalPolynomial(x, R) : RngPadElt, RngPad -> RngUPolElt
- CharacteristicPolynomial(x) : RngPadElt -> RngUPolElt
- CharacteristicPolynomial(x, R) : RngPadElt, RngPad -> RngUPolElt
- GaloisImage(x, i) : RngPadElt, RngIntElt -> RngPadElt
- Example RngLoc_agm (H48E17)
- EuclideanNorm(x) : RngPadResElt -> RngIntElt
- Power Relation (Algebraic Dependency)
- Teichmüller Lifts
- Linear Algebra
- Roots of Elements
- SquareRoot(x) : RngPadElt -> RngPadElt
- IsSquare(x) : RngPadElt -> BoolElt, RngPadElt
- InverseSquareRoot(x) : RngPadElt -> RngPadElt
- InverseSquareRoot(x, y) : RngPadElt, RngPadElt -> RngPadElt
- Root(x, n) : RngPadElt, RngIntElt -> RngPadElt
- IsPower(x, n) : RngPadElt, RngIntElt -> BoolElt, RngPadElt
- InverseRoot(x, n) : RngPadElt, RngIntElt -> RngPadElt
- InverseRoot(x, y, n) : RngPadElt, RngPadElt, RngIntElt -> RngPadElt
- Polynomials
- Automorphisms of Local Rings and Fields
- Automorphisms(L) : RngPad -> [Map]
- Automorphisms(K, k) : FldPad, FldPad -> [Map]
- AutomorphismGroup(L) : RngPad -> GrpPerm, Map
- AutomorphismGroup(K, k) : RngPad, RngPad -> GrpPerm, Map
- IsNormal(K) : RngPad -> BoolElt
- IsNormal(K, k) : RngPad, RngPad -> BoolElt
- IsAbelian(K, k) : FldPad, FldPad -> BoolElt
- Continuations(m, L) : Map, RngPad -> [Map]
- IsIsomorphic(E, K) : RngPad, RngPad -> BooElt
- Example RngLoc_units-autos (H48E26)
- GaloisGroup(f) : RngUPolElt[FldPad] -> GrpPerm, SeqEnum, UserProgram
- Example RngLoc_rngloc-galoisgroup (H48E27)
- Completions
- Class Field Theory
- Unit Group
- Norm Group
- NormGroup(R, m) : FldPad, Map -> GrpAb, Map
- NormEquation(R, m, b) : FldPad, Map, RngElt -> BoolElt, RngElt
- NormEquation(m1, m2, G) : Map, Map, GrpAb -> GrpAb, Map
- Norm(m1, m2, G) : Map, Map, GrpAb -> GrpAb
- NormKernel(m1, m2) : Map, Map -> GrpAb
- Class Fields
- Extensions
- Exact p-Adic Rings
- Introduction
- Exact p-adic Rings and Fields
- Exact p-adic Elements
- Polynomials over Exact p-adic Rings and Fields
- Exact Polynomial Rings
- Polynomials
- R ! f : RngUPolXPad, Any -> RngUPolXPadElt
- BaseRing(f) : RngUPolXPadElt -> Rng
- CanChangeRing(f, R) : RngUPolXPadElt, Rng -> BoolElt, RngUPolXPadElt
- Degree(f) : RngUPolXPadElt -> RngIntElt
- WeakDegree(f) : RngUPolXPadElt -> RngIntElt
- Coefficient(f, i) : RngUPolXPadElt, RngIntElt -> RngElt
- ExactPolynomial(f) : RngUPolXPadElt -> RngUPolXPadElt
- Evaluate(f, x) : RngUPolXPadElt, Any -> RngAnyXPadElt
- Derivative(f, m) : RngUPolXPadElt, RngIntElt -> RngUPolXPadElt
- Discriminant(f) : RngUPolXPadElt -> RngAnyXPadElt
- Resultant(f, g) : RngUPolXPadElt, RngUPolXPadElt -> RngAnyXPadElt
- IsInertial(f) : RngUPolXPadElt -> BoolElt
- IsWeaklyZero(f) : StrAnyXPadElt -> BoolElt
- IsDefinitelyZero(f) : StrAnyXPadElt -> BoolElt
- CoerceAndLift(S, x) : StrAnyXPad, Any -> StrAnyXPadElt
- Arithmetic
- Factorization and Roots
- NewtonPolygon(f) : RngUPolXPadElt[RngXPad] -> NwtnPgon
- Roots(f, R) : RngUPolElt, FldXPad -> SeqEnum
- HasRoot(f) : RngUPolXPadElt[RngXPad] -> BoolElt, RngXPadElt
- Factorization(f, R) : RngUPolXPadElt, FldXPad -> SeqEnum, RngXPadElt, SeqEnum
- IsIrreducible(f) : RngUPolXPadElt[RngXPad] -> BoolElt, Rec
- IsHenselLiftable(f, x) : RngUPolElt, FldXPadElt -> BoolElt, FldXPadElt
- RamificationResidualPolynomial(f, face) : RngUPolElt[FldXPad], NwtnPgonFace -> RngUPolElt
- RamificationResidualPolynomials(f) : RngUPolElt[FldXPad] -> SeqEnum, NwtnPgon
- Example RngLoc_fact-ex (H48E37)
- Bibliography
V2.28, 13 July 2023