p-ADIC RINGS AND FIELDS

DETAILS

 
Introduction

      p-adic Rings

      p-adic Fields

      p-adic Rings and Fields with Infinite Precision

      Definitions

 
Creation of p-adic Rings and Fields

      Creation Functions
            pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc

      Between Ring and Field
            IntegerRing(P) : FldLoc -> RngLoc
            FieldOfFractions(P) : RngLoc -> FldLoc

      Attributes of p-adic Rings and Fields
            P`DefaultPrecision
            P`SeriesPrinting
            Example RngPad_series-print (H42E1)

 
Elementary Invariants
      Prime(P) : FldLoc -> RngIntElt
      ResidueClassField(P) : RngLoc -> FldFin, Map
      Precision(P) : RngLoc -> RngIntElt
      Example RngPad_elinvar (H42E2)

 
Operations on Structures
      AssignNames(~P, S) : RngLoc, SeqEnum ->
      Characteristic(P) : RngLoc -> RngIntElt
      #P : RngLoc -> RngIntElt
      Name(P, k) : RngLoc, RngIntElt -> RngLocElt
      ChangePrecision(P, r) : RngLoc, RngIntElt -> RngLoc
      P1 eq P2 : RngLoc, RngLoc -> BoolElt
      P1 ne P2 : RngLoc, RngLoc -> BoolElt
      Example RngPad_strop (H42E3)

 
Element Constructions and Conversions

      Constructions
            elt<P | u> : RngLoc, RngElt -> RngLocElt
            elt<P | u, r> : RngLoc, RngElt, RngIntElt -> RngLocElt
            elt<P | v, u> : RngLoc, RngIntElt, RngElt -> RngLocElt
            elt<P | v, u, r> : RngLoc, RngIntElt, RngElt, RngIntElt -> RngLocElt
            BigO(x) : RngLocElt -> RngLocElt
            Example RngPad_eltcons (H42E4)

      Sequence Conversions
            ElementToSequence(x) : RngLocElt -> [ RngIntElt ]
            Coefficients(x) : RngLocElt -> [ RngLocElt ]
            Coefficient(x, i) : RngLocElt, RngIntElt -> RngLocElt
            Example RngPad_gal-desc (H42E5)

 
Operations on Elements

      Arithmetic
            Example RngPad_Division (H42E6)

      Equality and Membership
            Example RngPad_unram-ext (H42E7)

      Properties

      Precision and Valuation
            Example RngPad_ofe (H42E8)

      Logarithms and Exponentials
            Log(x) : RngLocElt -> RngLocElt
            Exp(x) : RngLocElt -> RngLocElt
            Example RngPad_log (H42E9)

 
Linear Algebra
      ManhattanForm(M) : Mtrx -> Mtrx, AlgMatElt
      Example RngPad_ech-man (H42E10)

 
Roots of Elements

 
Polynomials

      Operations for Polynomials
            GreatestCommonDivisor(g, h) : RngUPolElt, RngUPolElt -> RngUPolElt
            Example RngPad_gcd (H42E11)

      Roots of Polynomials

            Hensel Lifting of Roots
                  NewtonPolygon(g) : RngUPolElt -> NwtnPgon
                  ValuationsOfRoots(g) : RngUPolElt -> SeqEnum[<FldRatElt, RngIntElt>]
                  Example RngPad_newton-polygon (H42E12)
                  HenselLift(g, r) : RngUPolElt, RngLocElt -> RngLocElt
                  Example RngPad_Hensel (H42E13)

            Functions returning roots
                  Roots(g) : RngUPolElt -> [ <RngLocElt, RngIntElt> ]
                  HasRoot(g) : RngUPolElt -> BoolElt, RngLocElt
                  Example RngPad_ramified-ext (H42E14)

      Factorization
            HenselLift(f, g, h) : RngUPolElt, RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt
            Example RngPad_Poly-Hensel (H42E15)
            IsIrreducible(g) : RngUPolElt -> BoolElt
            SquareFreeFactorization(g) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
            Factorization(g) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
            SuggestedPrecision(f) : RngUPolElt -> RngIntElt
            Example RngPad_factors-infinite (H42E16)
            Example RngPad_factors-precision (H42E17)

 
Bibliography