Introduction

 
Background

 
Overview of the p-adics in Magma

      p-adic Rings

      p-adic Fields

      Free Precision Rings and Fields

      Precision of Extensions

 
Creation of Local Rings and Fields

      Creation Functions for the p-adics
            pAdicRing(p, k) : RngIntElt, RngIntElt -> RngPad
            pAdicRing(p) : RngIntElt -> RngPad
            pAdicQuotientRing(p, k) : RngIntElt, RngIntElt -> RngPadRes
            quo<L | x> : RngPad, RngPadElt -> .
            Example RngLoc_el_creation_padic (H49E1)

      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 (H49E2)

      Creation of Totally Ramified Extensions
            TotallyRamifiedExtension(L, f) : RngPad, RngUPolElt -> RngPad
            IsEisenstein(f) : RngUPolElt -> BoolElt
            Example RngLoc_el_creation_ram (H49E3)

      Creation of Unbounded Precision Extensions
            ext<L | m> : RngPad, Map -> RngPad
            Example RngLoc_el_creation_map (H49E4)

      Creation of Related Rings
            IntegerRing(F) : FldPad -> RngPad
            RingOfIntegers(R) : RngPad -> RngPad
            FieldOfFractions(R) : RngPad -> FldPad
            SplittingField(f, R) : RngUPolElt[RngInt], RngPad -> RngPad
            AbsoluteTotallyRamifiedExtension(R) : RngPad -> RngPad, Map

      Other Elementary Constructions
            Composite(R, S) : RngPad, RngPad -> RngPad

      Attributes of Local Rings and Fields
            L`DefaultPrecision : RngPad -> RngIntElt
            L`SeriesPrinting : RngPad -> BoolElt
            Example RngLoc_series_printing (H49E5)

 
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 (H49E6)
      AbsoluteRootNumber(K) : FldPad -> FldCycElt
      RootNumber(K) : FldPad -> FldCycElt
      Example RngLoc_padic-rootno-ex (H49E7)

 
Operations on Structures
      AssignNames(~L, S) : RngPad, SeqEnum ->
      Characteristic(L) : RngPad -> RngIntElt
      # L : RngPad -> RngIntElt
      Name(L, k) : RngPad, RngIntElt -> RngPadElt
      ChangePrecision(L, k) : RngPad, Any -> RngPad
      L eq K : RngPad, RngPad -> BoolElt
      L ne K : RngPad, RngPad -> BoolElt
      Example RngLoc_strop (H49E8)

      Ramification Predicates
            IsRamified(R) : RngPad -> BoolElt
            IsTamelyRamified(R) : RngPad -> BoolElt

 
Element Constructions and Conversions

      Constructions
            Zero(L) : RngPad -> RngPadElt
            One(L) : RngPad -> RngPadElt
            Random(L) : RngPad -> RngPadElt
            Representative(L) : RngPad -> RngPadElt
            elt<L | u> : RngPad, RngElt -> RngPadElt
            elt<L | u, r> : RngPad, RngElt, RngIntElt -> RngPadElt
            elt<L | v, u, r> : RngPad, RngIntElt, RngElt, RngIntElt -> RngPadElt
            BigO(x) : RngPadElt -> RngPadElt
            UniformizingElement(L) : RngPad -> RngPadElt
            Example RngLoc_eltcons (H49E9)
            Example RngLoc_eltcons_seq_weird (H49E10)

      Element Decomposers
            ElementToSequence(x) : RngPadElt -> [ RngElt ]
            Coefficient(x, i) : RngPadElt, RngIntElt -> RngPadElt
            Example RngLoc_gal-desc (H49E11)

 
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 (H49E12)

      Equality and Membership
            x eq y : RngPadResElt, RngPadResElt -> BoolElt
            x ne y : RngPadResElt, RngPadResElt -> BoolElt
            x in L : ., RngPad -> BoolElt
            x notin L : ., RngPad -> BoolElt
            Example RngLoc_unram-ext (H49E13)

      Properties
            IsZero(x) : RngPadElt -> BoolElt
            IsOne(x) : RngPadElt -> BoolElt
            IsMinusOne(x) : RngPadElt -> BoolElt
            IsUnit(x) : RngPadElt -> BoolElt
            IsIntegral(x) : RngPadElt -> BoolElt

      Precision and Valuation
            Parent(x) : RngPadElt -> RngPad
            Precision(x) : RngPadElt -> RngIntElt
            AbsolutePrecision(x) : RngPadElt -> RngIntElt
            RelativePrecision(x) : RngPadElt -> RngIntElt
            ChangePrecision(x, k) : RngUPolElt, RngIntElt -> RngPadElt
            Expand(x) : RngPadElt -> RngPadElt
            Valuation(x) : RngPadElt -> RngIntElt
            Example RngLoc_ofe (H49E14)
            Example RngLoc_padic-precision-woes (H49E15)

      Logarithms and Exponentials
            Log(x) : RngPadElt -> RngPadElt
            Exp(x) : RngPadElt -> RngPadElt
            Example RngLoc_log (H49E16)

      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 (H49E17)
            EuclideanNorm(x) : RngPadResElt -> RngIntElt

      Power Relation (Algebraic Dependency)
            PowerRelation(x,n) : FldPadElt, RngIntElt -> RngUPolElt

      Teichmüller Lifts
            TeichmuellerLift(u, R) : FldFinElt, RngPadResExt -> RngPadResExtElt

 
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

      Operations for Polynomials
            GreatestCommonDivisor(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
            Example RngLoc_gcd (H49E18)
            ShiftValuation(f, n) : RngUPolElt, RngIntElt -> RngUPolElt

      Roots of Polynomials

            Hensel Lifting of Roots
                  NewtonPolygon(f) : RngUPolElt -> NwtnPgon
                  ValuationsOfRoots(f) : RngUPolElt -> SeqEnum[<FldRatElt, RngIntElt>]
                  Example RngLoc_newton-polygon (H49E19)
                  HenselLift(f, x) : RngUPolElt, RngPadElt -> RngPadElt
                  Example RngLoc_Hensel (H49E20)

            Functions returning Roots
                  Roots(f) : RngUPolElt -> [ <RngPadElt, RngIntElt> ]
                  HasRoot(f) : RngUPolElt -> BoolElt, RngPadElt
                  Example RngLoc_ramified-ext (H49E21)

      Factorization
            HenselLift(f, s) : RngUPolElt, [RngUPolElt] -> [RngUPolElt]
            Example RngLoc_Poly-Hensel (H49E22)
            IsIrreducible(f) : RngUPolElt -> BoolElt
            SquareFreeFactorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
            Factorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
            SuggestedPrecision(f) : RngUPolElt -> RngIntElt
            IsIsomorphic(f, g) : RngUPolElt, RngUPolElt -> BoolElt
            Distance(f, g) : RngUPolElt, RngUPolElt -> RngIntElt
            Example RngLoc_factors-precision (H49E23)
            Example RngLoc_Factors (H49E24)
            SplittingField(f) : RngUPolElt[FldPad] -> FldPad, SeqEnum
            Example RngLoc_rngloc-splittingfield (H49E25)

 
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 (H49E26)
      GaloisGroup(f) : RngUPolElt[FldPad] -> GrpPerm, SeqEnum, UserProgram
      Example RngLoc_rngloc-galoisgroup (H49E27)

 
Completions
      Completion(O, P) : RngOrd, RngOrdIdl -> RngPad, Map
      LocalRing(P, k) : RngOrdIdl, RngIntElt -> RngPad, Map
      Example RngLoc_completion (H49E28)

 
Class Field Theory

      Unit Group
            PrincipalUnitGroupGenerators(R) : RngPad -> SeqEnum
            PrincipalUnitGroup(R) : RngPad -> GrpAb, Map
            UnitGroup(R) : RngPad -> GrpAb, Map
            UnitGroup(F) : FldPad -> GrpAb, Map
            UnitGroupGenerators(R) : RngPad -> SeqEnum
            UnitGroupGenerators(F) : FldPad -> SeqEnum
            pSelmerGroup(p,F) : RngIntElt, FldPad -> GrpAb, Map

      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
            ClassField(m, G) : Map, GrpAb -> FldAb
            NormGroupDiscriminant(m, G) : Map, GrpAb -> RngIntElt

 
Extensions
      AllExtensions(R, n) : RngPad, RngIntElt -> [RngPad]
      NumberOfExtensions(R, n) : RngPad, RngIntElt -> RngIntElt
      OreConditions(R, n, j) : RngPad, RngIntElt, RngIntElt -> BoolElt
      Example RngLoc_all-extensions (H49E29)

 
Exact p-Adic Rings

      Introduction
            IsExactpAdic(x) : Any -> BoolElt

      Exact p-adic Rings and Fields

            Construction of Exact p-adic Rings and Fields
                  pAdicField(p : parameters) : RngIntElt -> FldXPad
                  pAdicRing(p : parameters) : RngIntElt -> FldXPad
                  ext<K | f> : FldXPad, RngUPolElt -> RngXPad
                  Example RngLoc_constr_ex (H49E30)

            Related Structures
                  RingOfIntegers(L) : FldXPad -> RngXPad
                  FieldOfFractions(L) : FldXPad -> FldXPad
                  BaseField(F) : FldXPad -> FldXPad
                  BaseRing(R) : RngXPad -> RngXPad
                  ResidueClassField(L) : FldXPad -> FldFin, Map
                  pAdicQuotientRing(L, k) : FldXPad, RngIntElt -> RngPadRes, Map
                  InfinitePrecisionApproximation(K) : FldXPad -> FldPad
                  R eq T : RngXPad, RngXPad -> BoolElt
                  Example RngLoc_related-ex (H49E31)

            Generating Elements
                  R . i : FldXPad, RngIntElt -> FldXPadElt
                  AssignNames(~R, S) : RngXPad, SeqEnum[MonStgElt] ->
                  Generator(R) : FldXPad -> FldXPadElt
                  UniformizingElement(R) : FldXPad -> FldXPadElt
                  ResidueGenerator(R) : FldXPad -> FldXPadElt
                  AbsoluteGenerator(R) : FldXPad -> FldXPadElt
                  Example RngLoc_gen-ex (H49E32)

            Invariants
                  Prime(L) : FldXPad -> RngIntElt
                  Degree(L, K) : FldXPad, FldXPad -> RngIntElt
                  InertiaDegree(L, K) : FldXPad, FldXPad -> RngIntElt
                  RamificationDegree(L, K) : FldXPad, FldXPad -> RngIntElt
                  DefiningPolynomial(R) : RngXPad -> RngUPolElt
                  AbsoluteDegree(F) : FldXPad -> RngIntElt
                  DiscriminantValuation(L) : FldXPad -> RngIntElt
                  RamificationPolygon(L) : FldXPad -> NwtnPgon
                  Example RngLoc_invar-ex (H49E33)

      Exact p-adic Elements
            GetExactpAdicsPrintPrecision() : -> RngIntElt
            SetExactpAdicsPrintPrecision(k) : Infty ->
            K ! x : FldXPad, Any -> FldXPadElt
            AbsolutePrecision(x) : FldXPadElt -> RngIntElt
            RelativePrecision(x) : FldXPadElt -> RngIntElt
            Valuation(x) : FldXPadElt -> RngIntElt
            WeakValuation(x) : RngXPadElt -> RngIntElt
            ShiftValuation(x, n) : FldXPadElt, RngIntElt -> FldXPadElt
            ValuationEq(x, n) : FldXPadElt, RngIntElt -> BoolElt
            IsUnit(x) : RngXPadElt -> BoolElt
            IsIntegral(x) : RngXPadElt -> BoolElt
            IsWeaklyZero(x) : StrAnyXPadElt -> BoolElt
            IsDefinitelyZero(x) : StrAnyXPadElt -> BoolElt
            CoerceAndLift(S, x) : StrAnyXPad, Any -> StrAnyXPadElt

            Arithmetic with Elements
                  GCD(x, y) : FldXPadElt, FldXPadElt -> FldXPadElt
                  ExtendedGreatestCommonDivisor(x, y) : FldXPadElt, FldXPadElt -> FldXPadElt, FldXPadElt, FldXPadElt
                  Example RngLoc_elts-ex (H49E34)

      Polynomials over Exact p-adic Rings and Fields

            Exact Polynomial Rings
                  PolynomialRing(R : parameters) : Rng -> RngUPol
                  R eq T : RngUPolXPad, RngUPolXPad -> BoolElt
                  BaseRing(R) : RngUPolXPad -> Rng
                  R . i : RngUPolXPad, RngIntElt -> RngXPadElt
                  AssignNames(~R, S) : RngUPolXPad, SeqEnum[MonStgElt] ->
                  Example RngLoc_poly-ring-ex (H49E35)

            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
                  Example RngLoc_poly-ex (H49E36)

            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 (H49E37)

 
Bibliography

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