General \(p\)-Adic Extensions

• Introduction
• Constructions
  • LocalField(L, f): FldPad, RngUPolElt → RngLocA
  • LocalField(L, f): RngLocA, RngUPolElt → RngLocA
  • Example: One-Step Extension from a Two-Step Field
  • sub< L | a₁, ..., aₙ >: RngLocA, RngLocAElt, ..., RngLocAElt → RngLocA
  • sub< L | S>: RngLocA, SeqEnum → RngLocA
  • Example: Subfield Generated by an Element
• Operations with Fields
  • BaseRing(L): RngLocA → Rng
  • CoefficientRing(L): RngLocA → Rng
  • DefiningPolynomial(L): RngLocA → RngUPolElt
  • Degree(L): RngLocA → RngIntElt
  • Degree(L, R): RngLocA, Rng → RngIntElt
  • InertiaDegree(L): RngLocA → RngIntElt
  • RamificationDegree(L): RngLocA → RngIntElt
  • RamificationIndex(L): RngLocA → RngIntElt
  • Precision(L): RngLocA → RngIntElt
  • Prime(L): RngLocA → RngElt
  • Example: Coefficient Ring and Defining Polynomial
  • QuotientRepresentation(L): RngLocA → RngUPolRes
  • RamifiedRepresentation(L): RngLocA → FldPad, Map
  • Example: Two-Step Representation of a Local Field
  • 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
    • IsWildlyRamified(L): RngLocA → BoolElt
    • IsTotallyRamified(L): RngLocA → BoolElt
    • IsUnramified(L): RngLocA → BoolElt
• Maximal Order
  • IntegralBasis(L): RngLocA → SeqEnum
  • IsIntegral(a): RngLocAElt → BoolElt, SeqEnum
  • Example: Integral Basis of a Local Field
• Homomorphisms
  • hom< L -> R | a >: RngLocA, Rng, RngElt → Map
  • hom< L -> R | cfm, a >: RngLocA, Rng, Map, RngElt → Map
• Automorphisms and Galois Theory
  • FrobeniusAutomorphism(L): RngLocA → Map
  • AutomorphismGroup(L): RngLocA → Grp, Map
  • DecompositionGroup(L): RngLocA → GrpPerm
  • InertiaGroup(L): RngLocA → GrpPerm
  • RamificationGroup(L, i): RngLocA, RngIntElt → GrpPerm
  • FixedField(L, G): RngLocA, GrpPerm → RngLocA
  • Example: Automorphisms and Inertia Groups
• Elements Operations
  • L ! r: RngLocA, Any → RngLocAElt
  • L . i: RngLocA, RngIntElt → RngLocAElt
  • InertialElement(L): RngLocA → RngLocAElt
  • UniformizingElement(L): RngLocA → RngLocAElt
  • Arithmetic
    • a * b: RngLocAElt, RngLocAElt → RngLocAElt
    • a + b: RngLocAElt, RngLocAElt → RngLocAElt
    • a - b: RngLocAElt, RngLocAElt → RngLocAElt
    • - a: RngLocAElt → RngLocAElt
    • a ^ n: RngLocAElt, RngIntElt → RngLocAElt
    • a / b: RngLocAElt, RngLocAElt → RngLocAElt
  • Predicates on Elements
    • a eq b: RngLocAElt, RngLocAElt → BoolElt
    • IsOne(a): RngLocAElt → BoolElt
    • IsMinusOne(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
    • Norm(a): RngLocAElt → RngElt
    • Norm(a, F): RngLocAElt, Rng → RngElt
    • Trace(a): RngLocAElt → RngElt
    • Trace(a, F): RngLocAElt, Rng → RngElt
    • MinimalPolynomial(a): RngLocAElt → RngUPolElt
    • MinimalPolynomial(a, F): RngLocAElt → RngUPolElt
    • Example: Uniformizing and Inertial Elements
• Polynomial Factorization
  • Factorization(f): RngUPolElt[RngLocA] → SeqEnum, RngElt, Any
  • SuggestedPrecision(f): RngUPolElt[RngLocA] → RngIntElt
  • Roots(f): RngUPolElt[RngLocA] → SeqEnum
  • Roots(f, R): RngUPolElt, RngLocA → SeqEnum
  • Example: Factorization over a Local Field