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p-ADIC RINGS AND THEIR EXTENSIONS

 
Acknowledgements
 
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
      Creation Functions for Unramified Extensions
      Creation Functions for Totally Ramified Extensions
      Creation Functions for Unbounded Precision Extensions
      Miscellaneous Creation Functions
      Other Elementary Constructions
      Attributes of Local Rings and Fields
 
Elementary Invariants
 
Operations on Structures
      Ramification Predicates
 
Element Constructions and Conversions
      Constructions
      Element Decomposers
 
Operations on Elements
      Arithmetic
      Equality and Membership
      Properties
      Precision and Valuation
      Logarithms and Exponentials
      Norm and Trace Functions
      Teichmüller Lifts
 
Linear Algebra
 
Roots of Elements
 
Polynomials
      Operations for Polynomials
      Roots of Polynomials
            Hensel Lifting of Roots
            Functions returning Roots
      Factorization
 
Automorphisms of Local Rings and Fields
 
Completions
 
Class Field Theory
      Unit Group
      Norm Group
      Class Fields
 
Extensions
 
Bibliography







DETAILS

 
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 (H47E1)

      Creation Functions for 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 (H47E2)

      Creation Functions for Totally Ramified Extensions
            TotallyRamifiedExtension(L, f) : RngPad, RngUPolElt -> RngPad
            IsEisenstein(f) : RngUPolElt -> BoolElt
            Example RngLoc_el_creation_ram (H47E3)

      Creation Functions for Unbounded Precision Extensions
            ext<L | m> : RngPad, Map -> RngPad
            Example RngLoc_el_creation_map (H47E4)

      Miscellaneous Creation Functions
            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

 
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
      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 (H47E5)

 
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 (H47E6)

      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 (H47E7)
            Example RngLoc_eltcons_seq_weird (H47E8)

      Element Decomposers
            ElementToSequence(x) : RngPadElt -> [ RngElt ]
            Coefficient(x, i) : RngPadElt, RngIntElt -> RngPadElt
            Example RngLoc_gal-desc (H47E9)

 
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 (H47E10)

      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 (H47E11)

      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 (H47E12)

      Logarithms and Exponentials
            Log(x) : RngPadElt -> RngPadElt
            Exp(x) : RngPadElt -> RngPadElt
            Example RngLoc_log (H47E13)

      Norm and Trace Functions
            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 (H47E14)

      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 (H47E15)
            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 (H47E16)
                  HenselLift(f, x) : RngUPolElt, RngPadElt -> RngPadElt
                  Example RngLoc_Hensel (H47E17)

            Functions returning Roots
                  Roots(f) : RngUPolElt -> [ <RngPadElt, RngIntElt> ]
                  HasRoot(f) : RngUPolElt -> BoolElt, RngPadElt
                  Example RngLoc_ramified-ext (H47E18)

      Factorization
            HenselLift(f, s) : RngUPolElt, [RngUPolElt] -> [RngUPolElt]
            Example RngLoc_Poly-Hensel (H47E19)
            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 (H47E20)
            Example RngLoc_Factors (H47E21)

 
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 (H47E22)

 
Completions
      Completion(O, P) : RngOrd, RngOrdIdl -> RngPad, Map
      LocalRing(P, k) : RngOrdIdl, RngIntElt -> RngPad, Map
      Example RngLoc_completion (H47E23)

 
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

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013