Valuation Rings

• Introduction
• Creation Functions
  • Creation of Structures
    • ValuationRing(Q, p): FldRat, RngIntElt → RngVal
    • ValuationRing(F, f): FldFunRat, RngUPolElt → RngVal
    • ValuationRing(F): FldFunRat → RngVal
    • ValuationRing(K, p): FldNum, RngOrdIdl → RngVal
    • ValuationRing(K, p): FldFun, RngFunOrdIdl → RngVal
  • Creation of Elements
    • V ! r: RngVal, FldFunElt → RngValElt
    • V ! r: RngVal, FldRatElt → RngValElt
• Structure Operations
  • Related Structures
    • Category(V): RngVal → Cat
    • Parent(V): RngVal → Pow
    • PrimeRing(V): RngVal → RngVal
    • Center(V): RngVal → RngVal
    • FieldOfFractions(V): RngVal → Rng
  • Numerical Invariants
    • Characteristic(V): RngVal → RngIntElt
• Element Operations
  • Arithmetic Operations
    • + v: RngValElt → RngValElt
    • - v: RngValElt → RngValElt
    • v + w: RngValElt, RngValElt → RngValElt
    • v - w: RngValElt, RngValElt → RngValElt
    • v * w: RngValElt, RngValElt → RngValElt
    • v ^ k: RngValElt, RngIntElt → RngValElt
    • v / w: RngValElt, RngValElt → RngValElt
    • v +:= w: RngValElt, RngValElt → RngValElt
    • v -:= w: RngValElt, RngValElt → RngValElt
    • v *:= w: RngValElt, RngValElt → RngValElt
    • v div w: RngValElt, RngValElt → RngValElt
  • Equality and Membership
    • v eq w: RngValElt, RngValElt → BoolElt
    • v ne w: RngValElt, RngValElt → BoolElt
    • v in V: RngValElt, Rng → BoolElt
    • v notin V: RngValElt, Rng → BoolElt
  • Parent and Category
    • Parent(v): RngValElt → RngVal
    • Category(v): RngValElt → Cat
  • Predicates on Ring Elements
    • IsZero(n): RngValElt → BoolElt
    • IsOne(n): RngValElt → BoolElt
    • IsMinusOne(n): RngValElt → BoolElt
    • IsNilpotent(n): RngValElt → BoolElt
    • IsIdempotent(n): RngValElt → BoolElt
    • IsUnit(n): RngValElt → BoolElt
    • IsZeroDivisor(n): RngValElt → BoolElt
    • IsRegular(n): RngVal → BoolElt
  • Other Element Functions
    • EuclideanNorm(v): RngValElt → RngIntElt
    • Valuation(v): RngValElt → RngIntElt
    • Quotrem(v, w): RngValElt, RngValElt → RngValElt, RngValElt
    • GreatestCommonDivisor(v, w): RngValElt, RngValElt → RngValElt
    • Gcd(v, w): RngValElt, RngValElt → RngValElt
    • GCD(v, w): RngValElt, RngValElt → RngValElt
    • ExtendedGreatestCommonDivisor(v, w): RngValElt, RngValElt → RngValElt, RngValElt, RngValElt
    • Xgcd(v, w): RngValElt, RngValElt → RngValElt, RngValElt, RngValElt
    • XGCD(v, w): RngValElt, RngValElt → RngValElt, RngValElt, RngValElt