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GENERAL LOCAL FIELDS

 
Acknowledgements
 
Introduction
 
Constructions
 
Operations with Fields
      Predicates on Fields
 
Maximal Order
 
Homomorphisms from Fields
 
Automorphisms and Galois Theory
 
Local Field Elements
      Arithmetic
      Predicates on Elements
      Other Operations on Elements
 
Polynomials over General Local Fields







DETAILS

 
Introduction

 
Constructions
      LocalField(L, f) : FldPad, RngUPolElt -> RngLocA
      Example RngLocA_construct (H51E1)
      sub< L | a1, ..., an > : RngLocA, RngLocAElt, ..., RngLocAElt -> RngLocA
      Example RngLocA_sub (H51E2)

 
Operations with Fields
      BaseRing(L) : RngLocA -> Rng
      DefiningPolynomial(L) : RngLocA -> RngUPolElt
      Degree(L) : RngLocA -> RngIntElt
      Degree(L, R) : RngLocA, Rng -> RngIntElt
      InertiaDegree(L) : RngLocA -> RngIntElt
      Precision(L) : RngLocA -> RngIntElt
      Prime(L) : RngLocA -> RngElt
      Example RngLocA_ops (H51E3)
      QuotientRepresentation(L) : RngLocA -> RngUPolRes
      RamifiedRepresentation(L) : RngLocA -> FldPad, Map
      Example RngLocA_reps (H51E4)
      AssignNames(~L, S) : RngLocA, SeqEnum[MonStgElt] ->
      Name(L, i) : RngLocA, RngIntElt -> RngLocAElt
      Discriminant(L) : RngLocA -> RngElt
      ResidueClassField(L) : RngLocA -> Rng, Map
      RelativeField(L, m) : RngLocA, Map -> RngLocA, Map, Map

      Predicates on Fields
            IsRamified(L) : RngLocA -> BoolElt
            IsTamelyRamified(L) : RngLocA -> BoolElt
            IsTotallyRamified(L) : RngLocA -> BoolElt
            IsUnramified(L) : RngLocA -> BoolElt

 
Maximal Order
      IntegralBasis(L) : RngLocA -> SeqEnum
      IsIntegral(a) : RngLocAElt -> BoolElt, SeqEnum
      Example RngLocA_max-order (H51E5)

 
Homomorphisms from Fields
      hom< L -> R | a > : RngLocA, Rng, RngElt -> Map

 
Automorphisms and Galois Theory
      FrobeniusAutomorphism(L) : RngLocA -> Map
      AutomorphismGroup(L) : RngLocA -> Grp, Map
      DecompositionGroup(L) : RngLocA -> GrpPerm
      FixedField(L, G) : RngLocA, GrpPerm -> RngLocA
      Example RngLocA_auto-gal (H51E6)

 
Local Field Elements
      L ! r : RngLocA, Any -> RngLocAElt
      L . i : RngLocA, RngIntElt -> RngLocAElt
      InertialElement(L) : RngLocA -> RngLocAElt
      UniformizingElement(L) : RngLocA -> RngLocAElt

      Arithmetic

      Predicates on Elements
            a eq b : RngLocAElt, RngLocAElt -> BoolElt
            IsOne(a) : RngLocAElt -> BoolElt
            IsWeaklyZero(a) : RngLocAElt -> BoolElt
            IsZero(a) : RngLocAElt -> BoolElt

      Other Operations on Elements
            Valuation(a) : RngLocAElt -> RngExtReElt
            RelativePrecision(a) : RngLocAElt -> RngExtReElt
            Eltseq(a) : RngLocAElt -> SeqEnum
            RepresentationMatrix(a) : RngLocAElt -> AlgMatElt
            Example RngLocA_elts (H51E7)

 
Polynomials over General Local Fields
      Factorization(f) : RngUPolElt[RngLocA] -> SeqEnum, RngElt, Any
      SuggestedPrecision(f) : RngUPolElt[RngLocA] -> RngIntElt
      Roots(f) : RngUPolElt[RngLocA] -> SeqEnum
      Example RngLocA_poly-fact (H51E8)

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012