\(p\)-Adic Rings and Their Extensions
- Introduction
- Background
- Overview of the \(p\)-adics in Magma
- Creation of Local Rings and Fields
- Creation Functions for the \(p\)-adics
- Creation of Unramified Extensions
UnramifiedExtension(L, n): RngPad, RngIntElt → RngPadUnramifiedExtension(L, n): FldPad, RngIntElt → FldPadUnramifiedExtension(L, n): RngPadRes, RngIntElt → RngPadResExtUnramifiedExtension(L, n): RngPadResExt, RngIntElt → RngPadResExtext<L | n>: RngPad, RngIntElt → RngPadext<L | n>: FldPad, RngIntElt → FldPadext<L | n>: RngPadRes, RngIntElt → RngPadResExtext<L | n>: RngPadResExt, RngIntElt → RngPadResExtUnramifiedQuotientRing(K, k): FldFin, RngIntElt → RngUnramifiedExtension(L, f): RngPad, RngUPolElt → RngPadUnramifiedExtension(L, f): FldPad, RngUPolElt → FldPadUnramifiedExtension(L, f): RngPadRes, RngUPolElt → RngPadResExtUnramifiedExtension(L, f): RngPadResExt, RngUPolElt → RngPadResExtext<L | f>: RngPad, RngUPolElt → RngPadext<L | f>: FldPad, RngUPolElt → FldPadext<L | f>: RngPadRes, RngUPolElt → RngPadResExtext<L | f>: RngPadResExt, RngUPolElt → RngPadResExtIsInertial(f): RngUPolElt → BoolEltHasGNB(R, n, t): RngPad, RngIntElt, RngIntElt → BoolEltHasGNB(L, n, t): FldPad, RngIntElt, RngIntElt → BoolEltHasGNB(R, n, t): RngPadRes, RngIntElt, RngIntElt → BoolEltHasGNB(R, n, t): RngPadResExt, RngIntElt, RngIntElt → BoolEltCyclotomicUnramifiedExtension(R, f): FldPad, RngIntElt → FldPadCyclotomicUnramifiedExtension(R, f): RngPad, RngIntElt → RngPadCyclotomicUnramifiedExtension(R, f): RngPadRes, RngIntElt → RngPadResCyclotomicUnramifiedExtension(R, f): RngPadResExt, RngIntElt → RngPadResExt- Example: Creating Unramified Extensions
- Creation of Totally Ramified Extensions
TotallyRamifiedExtension(L, f): RngPad, RngUPolElt → RngPadTotallyRamifiedExtension(L, f): FldPad, RngUPolElt → FldPadTotallyRamifiedExtension(L, f): RngPadRes, RngUPolElt → RngPadResExtTotallyRamifiedExtension(L, f): RngPadResExt, RngUPolElt → RngPadResExtext<L | f>: RngPad, RngUPolElt → RngPadext<L | f>: FldPad, RngUPolElt → FldPadext<L | f>: RngPadRes, RngUPolElt → RngPadResExtext<L | f>: RngPadResExt, RngUPolElt → RngPadResExtIsEisenstein(f): RngUPolElt → BoolElt- Example: Creating Totally Ramified Extensions
- Creation of Unbounded Precision Extensions
- Creation of Related Rings
- Other Elementary Constructions
- Attributes of Local Rings and Fields
- Elementary Invariants
Prime(L): RngPad → RngIntEltPrime(L): FldPad → RngIntEltPrime(L): RngPadRes → RngIntEltPrime(L): RngPadResExt → RngIntEltInertiaDegree(L): RngPad → RngIntEltInertiaDegree(L): FldPad → RngIntEltInertiaDegree(L): RngPadRes → RngIntEltInertiaDegree(L): RngPadResExt → RngIntEltInertiaDegree(K, L): RngPad, RngPad → RngIntEltInertiaDegree(K, L): FldPad, FldPad → RngIntEltInertiaDegree(K, L): RngPadRes, RngPadRes → RngIntEltInertiaDegree(K, L): RngPadResExt, RngPadResExt → RngIntEltAbsoluteInertiaDegree(L): RngPad → RngIntEltAbsoluteInertiaDegree(L): FldPad → RngIntEltAbsoluteInertiaDegree(L): RngPadRes → RngIntEltAbsoluteInertiaDegree(L): RngPadResExt → RngIntEltAbsoluteInertiaIndex(L): RngPad → RngIntEltAbsoluteInertiaIndex(L): FldPad → RngIntEltAbsoluteInertiaIndex(L): RngPadRes → RngIntEltAbsoluteInertiaIndex(L): RngPadResExt → RngIntEltRamificationDegree(L): RngPad → RngIntEltRamificationDegree(L): FldPad → RngIntEltRamificationDegree(L): RngPadRes → RngIntEltRamificationDegree(L): RngPadResExt → RngIntEltRamificationIndex(L): RngPad → RngIntEltRamificationIndex(L): FldPad → RngIntEltRamificationIndex(L): RngPadRes → RngIntEltRamificationIndex(L): RngPadResExt → RngIntEltRamificationDegree(K, L): RngPad, RngPad → RngIntEltRamificationDegree(K, L): FldPad, FldPad → RngIntEltRamificationDegree(K, L): RngPadRes, RngPadRes → RngIntEltRamificationDegree(K, L): RngPadResExt, RngPadResExt → RngIntEltRamificationIndex(K, L): RngPad, RngPad → RngIntEltRamificationIndex(K, L): FldPad, FldPad → RngIntEltRamificationIndex(K, L): RngPadRes, RngPadRes → RngIntEltRamificationIndex(K, L): RngPadResExt, RngPadResExt → RngIntEltAbsoluteRamificationDegree(L): RngPad → RngIntEltAbsoluteRamificationDegree(L): FldPad → RngIntEltAbsoluteRamificationDegree(L): RngPadRes → RngIntEltAbsoluteRamificationDegree(L): RngPadResExt → RngIntEltAbsoluteRamificationIndex(L): RngPad → RngIntEltAbsoluteRamificationIndex(L): FldPad → RngIntEltAbsoluteRamificationIndex(L): RngPadRes → RngIntEltAbsoluteRamificationIndex(L): RngPadResExt → RngIntEltAbsoluteDegree(L): RngPad → RngIntEltDegree(L): RngPad → RngIntEltDegree(L): FldPad → RngIntEltDegree(L): RngPadRes → RngIntEltDegree(L): RngPadResExt → RngIntEltDegree(K, L): RngPad, RngPad → RngIntEltDegree(K, L): FldPad, FldPad → RngIntEltDegree(K, L): RngPadRes, RngPadRes → RngIntEltDegree(K, L): RngPadResExt, RngPadResExt → RngIntEltDefiningPolynomial(L): RngPad → RngUPolEltDefiningPolynomial(L): FldPad → RngUPolEltDefiningPolynomial(L): RngPadRes → RngUPolEltDefiningPolynomial(L): RngPadResExt → RngUPolEltDefiningPolynomial(K, L): RngPad, RngPad → RngUPolEltDefiningPolynomial(K, L): FldPad, FldPad → RngUPolEltDefiningPolynomial(K, L): RngPadRes, RngPadRes → RngUPolEltDefiningPolynomial(K, L): RngPadResExt, RngPadRes → RngUPolEltDefiningPolynomial(K, L): RngPadResExt, RngPadResExt → RngUPolEltDefiningMap(L): RngPad → MapDefiningMap(L): FldPad → MapHasDefiningMap(L): RngPad → BoolElt, MapDefiningMap(L): FldPad → BoolElt, MapPrimeRing(L): RngPad → RngPadPrimeRing(L): RngPadRes → RngPadResPrimeRing(L): RngPadResExt → RngPadResPrimeField(L): FldPad → FldPadpAdicRing(L): RngPad → RngPadpAdicRing(L): RngPadRes → RngPadRespAdicRing(L): RngPadResExt → RngPadRespAdicField(L): FldPad → FldPadBaseRing(L): RngPad → RngPadBaseRing(L): RngPadRes → RngPadResBaseRing(L): RngPadResExt → RngCoefficientRing(L): RngPad → RngPadCoefficientRing(L): RngPadRes → RngPadResCoefficientRing(L): RngPadResExt → RngBaseField(L): FldPad → FldPadCoefficientField(L): FldPad → FldPadBaseRing(L): FldPad → FldPadResidueClassField(L): RngPad → FldFin, MapResidueClassField(L): RngPadRes → FldFin, MapResidueClassField(L): RngPadResExt → FldFin, MapResidueSystem(R): RngPad → [RngPadElt]ResidueSystem(R): RngPadRes → [RngPadEltRes]ResidueSystem(R): RngPadResExt → [RngPadEltResExt]ResidueSystem(R): FldPad → [FldPadElt]UniformizingElement(L): RngPad → RngPadEltUniformizingElement(L): RngPadRes → RngPadResEltUniformizingElement(L): RngPadResExt → RngPadResExtEltUniformizingElement(L): FldPad → FldPadEltL . 1: RngPad → RngPadEltL . 1: RngPadRes → RngPadResEltL . 1: RngPadResExt → RngPadResExtEltL . 1: FldPad → FldPadEltPrecision(L): RngPad → RngIntEltPrecision(L): RngPadRes → RngIntEltPrecision(L): RngPadResExt → RngIntEltPrecision(L): FldPad → RngIntEltHasPRoot(R): RngPad → BoolEltHasRootOfUnity(L, n): RngPad, RngIntElt → BoolEltDiscriminant(R): RngPad → RngPadEltDiscriminant(K, k): RngPad, RngPad → RngPadEltAdditiveGroup(R): RngPadRes → GrpAb, Map- Example: Elementary Invariants of Local Rings
AbsoluteRootNumber(K): FldPad → FldCycEltRootNumber(K): FldPad → FldCycElt- Example: Root Numbers of p-adic Extensions
- Operations on Structures
AssignNames(~L, S): RngPad, SeqEnumAssignNames(~L, S): RngPadResExt, SeqEnumAssignNames(~L, S): FldPad, SeqEnumCharacteristic(L): RngPad → RngIntEltCharacteristic(L): RngPadRes → RngIntEltCharacteristic(L): RngPadResExt → RngIntEltCharacteristic(L): FldPad → RngIntElt# L: RngPad → RngIntEltName(L, k): RngPad, RngIntElt → RngPadEltName(L, k): RngPadRes, RngIntElt → RngPadResEltName(L, k): RngPadResExt, RngIntElt → RngPadResExtEltName(L, k): FldPad, RngIntElt → FldPadEltChangePrecision(L, k): RngPad, Any → RngPadChangePrecision(L, k): RngPad, Infty → RngPadChangePrecision(~L, k): RngPad, Infty → RngPadChangePrecision(L, k): RngPad, RngIntElt → RngPadChangePrecision(~L, k): RngPad, RngIntElt → RngPadChangePrecision(L, k): RngPadRes, RngIntElt → RngPadResChangePrecision(~L, k): RngPadRes, RngIntElt → RngPadResChangePrecision(L, k): RngPadResExt, RngIntElt → RngPadResExtChangePrecision(~L, k): RngPadResExt, RngIntElt → RngPadResExtChangePrecision(L, k): FldPad, RngIntElt → FldPadChangePrecision(~L, k): FldPad, RngIntElt → FldPadChangePrecision(L, k): FldPad, Any → FldPadChangePrecision(L, k): FldPad, Infty → FldPadChangePrecision(~L, k): FldPad, Infty → FldPadL eq K: RngPad, RngPad → BoolEltL eq K: RngPadRes, RngPadRes → BoolEltL eq K: RngPadResExt, RngPadResExt → BoolEltL eq K: FldPad, FldPad → BoolEltL ne K: RngPad, RngPad → BoolEltL ne K: RngPadRes, RngPadRes → BoolEltL ne K: RngPadResExt, RngPadResExt → BoolEltL ne K: FldPad, FldPad → BoolElt- Example: Changing Precision of Local Structures
- Ramification Predicates
- Element Constructions and Conversions
- Constructions
Zero(L): RngPad → RngPadEltZero(L): RngPadRes → RngPadResEltZero(L): RngPadResExt → RngPadResExtEltZero(L): FldPad → FldPadEltOne(L): RngPad → RngPadEltOne(L): RngPadRes → RngPadResEltOne(L): RngPadResExt → RngPadResExtEltOne(L): FldPad → FldPadEltRandom(L): RngPad → RngPadEltRandom(L): RngPadRes → RngPadResEltRandom(L): RngPadResExt → RngPadResExtEltRepresentative(L): RngPad → RngPadEltRepresentative(L): RngPadRes → RngPadResEltRepresentative(L): RngPadResExt → RngPadResExtEltRepresentative(L): FldPad → FldPadEltelt<L | u>: RngPad, RngElt → RngPadEltelt<L | u>: RngPad, [RngElt] → RngPadEltL ! u: RngPad, RngElt → RngPadEltL ! u: RngPad, [RngElt] → RngPadEltelt<L | u>: FldPad, RngElt → FldPadEltelt<L | u>: FldPad, [RngElt] → FldPadEltL ! u: FldPad, RngElt → FldPadEltL ! u: FldPad, [RngElt] → FldPadEltelt<L | u, r>: RngPad, RngElt, RngIntElt → RngPadEltelt<L | u, r>: RngPad, [RngElt], RngIntElt → RngPadEltelt<L | u, r>: FldPad, RngElt, RngIntElt → FldPadEltelt<L | u, r>: FldPad, [RngElt], RngIntElt → FldPadEltelt<L | v, u, r>: RngPad, RngIntElt, RngElt, RngIntElt → RngPadEltelt<L | v, u, r>: RngPad, RngIntElt, [RngElt], RngIntElt → RngPadEltelt<L | v, u, r>: FldPad, RngIntElt, RngElt, RngIntElt → FldPadEltelt<L | v, u, r>: FldPad, RngIntElt, [RngElt], RngIntElt → FldPadEltBigO(x): RngPadElt → RngPadEltBigO(x): FldPadElt → FldPadEltO(x): RngPadElt → RngPadEltO(x): FldPadElt → FldPadEltUniformizingElement(L): RngPad → RngPadEltUniformizingElement(L): RngPadRes → RndPadResEltUniformizingElement(L): RngPadResExt → RndPadResExtEltUniformizingElement(L): FldPad → FldPadElt- Example: Element Constructors and Imprecise Zeros
- Example: Precision When Coercing Sequences
- Element Decomposers
- Operations on Elements
- Arithmetic
- x: RngPadElt → RngPadElt- x: RngPadResElt → RngPadResElt- x: RngPadResExtElt → RngPadResExtElt- x: FldPadElt → FldPadEltx + y: RngPadElt, RngPadElt → RngPadEltx + y: RngPadResElt, RngPadResElt → RngPadResEltx + y: RngPadResExtElt, RngPadResExtElt → RngPadResExtEltx + y: FldPadElt, FldPadElt → FldPadEltx - y: RngPadElt, RngPadElt → RngPadEltx - y: RngPadResElt, RngPadResElt → RngPadResEltx - y: RngPadResExtElt, RngPadResExtElt → RngPadResExtEltx - y: FldPadElt, FldPadElt → FldPadEltx * y: RngPadElt, RngPadElt → RngPadEltx * y: RngPadResElt, RngPadResElt → RngPadResEltx * y: RngPadResExtElt, RngPadResExtElt → RngPadResExtEltx * y: FldPadElt, FldPadElt → FldPadEltx ^ k: RngPadElt, RngIntElt → RngPadEltx ^ k: RngPadResElt, RngIntElt → RngPadResEltx ^ k: RngPadResExtElt, RngIntElt → RngPadResExtEltx ^ k: FldPadElt, RngIntElt → FldPadEltx div y: RngPadElt, RngPadElt → RngPadEltx div y: RngPadResElt, RngPadResElt → RngPadResEltx div y: RngPadResExtElt, RngPadResExtElt → RngPadResExtEltx div y: FldPadElt, FldPadElt → FldPadEltx div:= y: RngPadElt, RngPadElt → RngPadEltx div:= y: RngPadResElt, RngPadResElt → RngPadResEltx div:= y: RngPadElt, RngPadElt → RngPadEltx div:= y: FldPadElt, FldPadElt → FldPadEltx / y: RngPadElt, RngPadElt → RngPadEltx / y: FldPadElt, FldPadElt → FldPadEltIsExactlyDivisible(x, y): RngPadElt, RngPadElt → BoolElt, RngPadEltIsExactlyDivisible(x, y): RngPadResElt, RngPadResElt → BoolElt, RngPadResEltIsExactlyDivisible(x, y): RngPadResExtElt, RngPadResExtElt → BoolElt, RngPadResExtEltIsExactlyDivisible(x, y): FldPadElt, FldPadElt → BoolElt, FldPadElt- Example: Division of Non-units
- Equality and Membership
x eq y: RngPadResElt, RngPadResElt → BoolEltx eq y: RngPadResExtElt, RngPadResExtElt → BoolEltx eq y: RngPadElt, RngPadElt → BoolEltx eq y: FldPadElt, FldPadElt → BoolEltx ne y: RngPadResElt, RngPadResElt → BoolEltx ne y: RngPadResExtElt, RngPadResExtElt → BoolEltx ne y: RngPadElt, RngPadElt → BoolEltx ne y: FldPadElt, FldPadElt → BoolEltx in L: ., RngPad → BoolEltx in L: ., FldPad → BoolEltx notin L: ., RngPad → BoolEltx notin L: ., FldPad → BoolElt- Example: Constructing an Unramified Extension
- Properties
- Precision and Valuation
Parent(x): RngPadElt → RngPadParent(x): RngPadResElt → RngPadResParent(x): RngPadResExtElt → RngPadResExtParent(x): FldPadElt → FldPadPrecision(x): RngPadElt → RngIntEltPrecision(x): RngPadResElt → RngIntEltPrecision(x): RngPadResExtElt → RngIntEltPrecision(x): FldPadElt → RngIntEltAbsolutePrecision(x): RngPadElt → RngIntEltAbsolutePrecision(x): RngPadResElt → RngIntEltAbsolutePrecision(x): RngPadResExtElt → RngIntEltAbsolutePrecision(x): FldPadElt → RngIntEltRelativePrecision(x): RngPadElt → RngIntEltRelativePrecision(x): RngPadResElt → RngIntEltRelativePrecision(x): RngPadResExtElt → RngIntEltRelativePrecision(x): FldPadElt → RngIntEltChangePrecision(x, k): RngUPolElt, RngIntElt → RngPadEltChangePrecision(~x, k): RngUPolElt, RngIntElt → RngPadEltChangePrecision(x, k): RngPadElt, RngIntElt → RngPadEltChangePrecision(~x, k): RngPadElt, RngIntElt → RngPadEltChangePrecision(x, k): FldPadElt, RngIntElt → FldPadEltChangePrecision(~x, k): FldPadElt, RngIntElt → FldPadEltExpand(x): RngPadElt → RngPadEltExpand(x): FldPadElt → FldPadEltValuation(x): RngPadElt → RngIntEltValuation(x): RngPadResElt → RngIntEltValuation(x): RngPadResExtElt → RngIntEltValuation(x): FldPadElt → RngIntElt- Example: Relative and Absolute Precision
- Example: Pitfalls of Unbounded Precision
- Logarithms and Exponentials
- Norm and Trace
Norm(x): RngPadElt → RngPadEltNorm(x): RngPadResElt → RngPadResEltNorm(x): RngPadResExtElt → RngEltNorm(x): FldPadElt → FldPadEltNorm(x, R): RngPadElt, RngPad → RngPadEltNorm(x, R): RngPadResElt, RngPadRes → RngPadResEltNorm(x, R): RngPadResExtElt, Rng → RngEltNorm(x, R): FldPadElt, FldPad → FldPadEltTrace(x): RngPadElt → RngPadEltTrace(x): RngPadResElt → RngPadResEltTrace(x): RngPadResExtElt → RngEltTrace(x): FldPadElt → FldPadEltTrace(x, R): RngPadElt, RngPad → RngPadEltTrace(x, R): RngPadResElt, RngPadRes → RngPadResEltTrace(x, R): RngPadResExtElt, Rng → RngEltTrace(x, R): FldPadElt, FldPad → FldPadEltMinimalPolynomial(x): RngPadElt → RngUPolEltMinimalPolynomial(x): RngPadResElt → RngUPolEltMinimalPolynomial(x): RngPadResExtElt → RngUPolEltMinimalPolynomial(x): FldPadElt → RngUPolEltMinimalPolynomial(x, R): RngPadElt, RngPad → RngUPolEltMinimalPolynomial(x, R): RngPadResElt, RngPadRes → RngUPolEltMinimalPolynomial(x, R): RngPadResExtElt, Rng → RngUPolEltMinimalPolynomial(x, R): FldPadElt, FldPad → RngUPolEltCharacteristicPolynomial(x): RngPadElt → RngUPolEltCharacteristicPolynomial(x): RngPadResElt → RngUPolEltCharacteristicPolynomial(x): RngPadResExtElt → RngUPolEltCharacteristicPolynomial(x, R): RngPadElt, RngPad → RngUPolEltCharacteristicPolynomial(x, R): RngPadResElt, RngPadRes → RngUPolEltCharacteristicPolynomial(x, R): RngPadResExtElt, RngPadRes → RngUPolEltCharacteristicPolynomial(x, R): RngPadResExtElt, RngPadResExt → RngUPolEltGaloisImage(x, i): RngPadElt, RngIntElt → RngPadEltGaloisImage(x, i): RngPadResElt, RngIntElt → RngPadResEltGaloisImage(x, i): RngPadResExtElt, RngIntElt → RngPadResExtEltGaloisImage(x, i): FldPadElt, RngIntElt → FldPadElt- Example: Arithmetic-Geometric Mean Point Counting
EuclideanNorm(x): RngPadResElt → RngIntEltEuclideanNorm(x): RngRadResExtElt → RngIntElt
- Power Relation (Algebraic Dependency)
- Teichmüller Lifts
- Linear Algebra
- Roots of Elements
SquareRoot(x): RngPadElt → RngPadEltSquareRoot(x): RngPadResElt → RngPadResEltSquareRoot(x): RngPadResExtElt → RngPadResExtEltSquareRoot(x): FldPadElt → FldPadEltSqrt(x): RngPadElt → RngPadEltSqrt(x): RngPadResElt → RngPadResEltSqrt(x): RngPadResExtElt → RngPadResExtEltSqrt(x): FldPadElt → FldPadEltIsSquare(x): RngPadElt → BoolElt, RngPadEltIsSquare(x): RngPadResElt → BoolElt, RngPadResEltIsSquare(x): RngPadResExtElt → BoolElt, RngPadResExtEltIsSquare(x): FldPadElt → BoolElt, FldPadEltInverseSquareRoot(x): RngPadElt → RngPadEltInverseSquareRoot(x): RngPadResElt → RngPadResEltInverseSquareRoot(x): RngPadResExtElt → RngPadResExtEltInverseSquareRoot(x): FldPadElt → FldPadEltInverseSqrt(x): RngPadElt → RngPadEltInverseSqrt(x): RngPadResElt → RngPadResEltInverseSqrt(x): RngPadResExtElt → RngPadResExtEltInverseSqrt(x): FldPadElt → FldPadEltInverseSquareRoot(x, y): RngPadElt, RngPadElt → RngPadEltInverseSquareRoot(x, y): RngPadResElt, RngPadResElt → RngPadResEltInverseSquareRoot(x, y): RngPadResExtElt, RngPadResExtElt → RngPadResExtEltInverseSquareRoot(x, y): FldPadElt, FldPadElt → FldPadEltInverseSqrt(x, y): RngPadElt, RngPadElt → RngPadEltInverseSqrt(x, y): RngPadResElt, RngPadResElt → RngPadResEltInverseSqrt(x, y): RngPadResExtElt, RngPadResExtElt → RngPadResExtEltInverseSqrt(x, y): FldPadElt, FldPadElt → FldPadEltRoot(x, n): RngPadElt, RngIntElt → RngPadEltRoot(x, n): RngPadResElt, RngIntElt → RngPadResEltRoot(x, n): RngPadResExtElt, RngIntElt → RngPadResExtEltRoot(x, n): FldPadElt, RngIntElt → FldPadEltIsPower(x, n): RngPadElt, RngIntElt → BoolElt, RngPadEltIsPower(x, n): RngPadResElt, RngIntElt → BoolElt, RngPadResEltIsPower(x, n): RngPadResExtElt, RngIntElt → BoolElt, RngPadResExtEltIsPower(x, n): FldPadElt, RngIntElt → BoolElt, FldPadEltInverseRoot(x, n): RngPadElt, RngIntElt → RngPadEltInverseRoot(x, n): RngPadResElt, RngIntElt → RngPadResEltInverseRoot(x, n): RngPadResExtElt, RngIntElt → RngPadResExtEltInverseRoot(x, n): FldPadElt, RngIntElt → FldPadEltInverseRoot(x, y, n): RngPadElt, RngPadElt, RngIntElt → RngPadEltInverseRoot(x, y, n): RngPadResElt, RngPadResElt, RngIntElt → RngPadResEltInverseRoot(x, y, n): RngPadResExtElt, RngPadResExtElt, RngIntElt → RngPadResExtEltInverseRoot(x, y, n): FldPadElt, FldPadElt, RngIntElt → FldPadElt
- Polynomials
- Operations for Polynomials
GreatestCommonDivisor(f, g): RngUPolElt, RngUPolElt → RngUPolEltGcd(f, g): RngUPolElt, RngUPolElt → RngUPolEltGCD(f, g): RngUPolElt, RngUPolElt → RngUPolEltf div g: RngUPolElt, RngUPolElt → RngUPolEltf mod g: RngUPolElt, RngUPolElt → RngUPolEltLeastCommonMultiple(f, g): RngUPolElt, RngUPolElt → RngUPolEltCoefficient(f, i): RngUPolElt, RngIntElt → RngEltLeadingCoefficient(f): RngUPolElt → RngEltDerivative(f): RngUPolElt → RngUPolEltEvaluate(f, x): RngUPolElt, RngElt → RngElt- Example: GCDs of p-adic Polynomials
ShiftValuation(f, n): RngUPolElt, RngIntElt → RngUPolElt
- Roots of Polynomials
- Hensel Lifting of Roots
NewtonPolygon(f): RngUPolElt → NwtnPgonValuationsOfRoots(f): RngUPolElt → SeqEnum[<FldRatElt, RngIntElt>]- Example: Newton Polygons and Root Valuations
HenselLift(f, x): RngUPolElt, RngPadElt → RngPadEltHenselLift(f, x): RngUPolElt, RngPadResElt → RngPadResEltHenselLift(f, x): RngUPolElt, RngPadResExtElt → RngPadResExtEltHenselLift(f, x): RngUPolElt, FldPadElt → FldPadEltHenselLift(f, x, k): RngUPolElt, RngPadElt, RngIntElt → RngPadEltHenselLift(f, x, k): RngUPolElt, RngPadResElt, RngIntElt → RngPadResEltHenselLift(f, x, k): RngUPolElt, RngPadResExtElt, RngIntElt → RngPadResExtEltHenselLift(f, x, k): RngUPolElt, FldPadElt, RngIntElt → FldPadElt- Example: Square Roots by Hensel Lifting
- Functions returning Roots
- Factorization
HenselLift(f, s): RngUPolElt, [RngUPolElt] → [RngUPolElt]- Example: Hensel Factorization of Polynomials
IsIrreducible(f): RngUPolElt → BoolEltSquareFreeFactorization(f): RngUPolElt → [ < RngUPolElt, RngIntElt > ]Factorization(f): RngUPolElt → [ < RngUPolElt, RngIntElt > ]LocalFactorization(f): RngUPolElt → [ < RngUPolElt, RngIntElt >]SuggestedPrecision(f): RngUPolElt → RngIntEltIsIsomorphic(f, g): RngUPolElt, RngUPolElt → BoolEltDistance(f, g): RngUPolElt, RngUPolElt → RngIntElt- Example: Suggested Precision for Factorization
- Example: Local Factorizations and Galois Groups
SplittingField(f): RngUPolElt[FldPad] → FldPad, SeqEnumSplittingField(f): RngUPolElt[RngPad] → FldPad, SeqEnum- Example: Splitting Field over a p-adic Field
- Automorphisms of Local Rings and Fields
Automorphisms(L): RngPad → [Map]Automorphisms(L): FldPad → [Map]Automorphisms(K, k): FldPad, FldPad → [Map]Automorphisms(K, k): RngPad, RngPad → [Map]AutomorphismGroup(L): RngPad → GrpPerm, MapAutomorphismGroup(L): FldPad → GrpPerm, MapAutomorphismGroup(K, k): RngPad, RngPad → GrpPerm, MapAutomorphismGroup(K, k): FldPad, FldPad → GrpPerm, MapIsNormal(K): RngPad → BoolEltIsNormal(K): FldPad → BoolEltIsNormal(K, k): RngPad, RngPad → BoolEltIsNormal(K, k): FldPad, FldPad → BoolEltIsAbelian(K, k): FldPad, FldPad → BoolEltContinuations(m, L): Map, RngPad → [Map]IsIsomorphic(E, K): RngPad, RngPad → BooEltIsIsomorphic(E, K): FldPad, FldPad → BooElt- Example: Automorphisms of a Local Ring
GaloisGroup(f): RngUPolElt[FldPad] → GrpPerm, SeqEnum, UserProgramGaloisGroup(f): RngUPolElt[RngPad] → GrpPerm, SeqEnum, UserProgram- Example: Galois Group of a Local Polynomial
- Completions
- Class Field Theory
- Unit Group
- Norm Group
NormGroup(R, m): FldPad, Map → GrpAb, MapNormGroup(R, m): RngPad, Map → GrpAb, MapNormEquation(R, m, b): FldPad, Map, RngElt → BoolElt, RngEltNormEquation(R, m, b): RngPad, Map, RngElt → BoolElt, RngEltNormEquation(m1, m2, G): Map, Map, GrpAb → GrpAb, MapNorm(m1, m2, G): Map, Map, GrpAb → GrpAbNormKernel(m1, m2): Map, Map → GrpAb
- Class Fields
- Extensions
AllExtensions(R, n): RngPad, RngIntElt → [RngPad]AllExtensions(R, n): FldPad, RngIntElt → [RngPad]NumberOfExtensions(R, n): RngPad, RngIntElt → RngIntEltOreConditions(R, n, j): RngPad, RngIntElt, RngIntElt → BoolEltOreConditions(R, n, j): FldPad, RngIntElt, RngIntElt → BoolElt- Example: All Extensions of Degree 9 over Q3
- Exact \(p\)-Adic Rings
- Introduction
- Exact \(p\)-adic Rings and Fields
- Construction of Exact \(p\)-adic Rings and Fields
- Related Structures
RingOfIntegers(L): FldXPad → RngXPadRingOfIntegers(L): RngXPad → RngXPadIntegerRing(L): FldXPad → RngXPadIntegers(L): FldXPad → RngXPadIntegerRing(L): RngXPad → RngXPadIntegers(L): RngXPad → RngXPadFieldOfFractions(L): FldXPad → FldXPadFieldOfFractions(L): RngXPad → FldXPadBaseField(F): FldXPad → FldXPadCoefficientField(F): FldXPad → FldXPadBaseRing(R): RngXPad → RngXPadCoefficientRing(R): RngXPad → RngXPadResidueClassField(L): FldXPad → FldFin, MapResidueClassField(L): RngXPad → FldFin, MappAdicQuotientRing(L, k): FldXPad, RngIntElt → RngPadRes, MappAdicQuotientRing(L, k): RngXPad, RngIntElt → RngPadRes, Mapquo<L | x>: RngXPad, Any → RngPadRes, Mapquo<L | x>: FldXPad, Any → RngPadRes, MapInfinitePrecisionApproximation(K): FldXPad → FldPadInfinitePrecisionApproximation(K): RngXPad → RngPadR eq T: RngXPad, RngXPad → BoolEltR eq T: FldXPad, FldXPad → BoolElt- Example: Structures Related to Exact p-adic Rings
- Generating Elements
R . i: FldXPad, RngIntElt → FldXPadEltR . i: RngXPad, RngIntElt → RngXPadEltName(R, i): RngXPad, RngIntElt → RngXPadEltName(R, i): FldXPad, RngIntElt → FldXPadEltAssignNames(~R, S): RngXPad, SeqEnum[MonStgElt]AssignNames(~R, S): FldXPad, SeqEnum[MonStgElt]Generator(R): FldXPad → FldXPadEltGenerator(R): RngXPad → RngXPadEltUniformizingElement(R): FldXPad → FldXPadEltUniformizingElement(R): RngXPad → RngXPadEltResidueGenerator(R): FldXPad → FldXPadEltResidueGenerator(R): RngXPad → RngXPadEltAbsoluteGenerator(R): FldXPad → FldXPadEltAbsoluteGenerator(R): RngXPad → RngXPadElt- Example: Generating Elements of Exact p-adic Rings
- Invariants
Prime(L): FldXPad → RngIntEltPrime(L): RngXPad → RngIntEltDegree(L, K): FldXPad, FldXPad → RngIntEltDegree(L): FldXPad → RngIntEltDegree(L): RngXPad → RngIntEltDegree(L, K): RngXPad, RngXPad → RngIntEltInertiaDegree(L, K): FldXPad, FldXPad → RngIntEltInertiaDegree(L): FldXPad → RngIntEltInertiaDegree(L, K): RngXPad, RngXPad → RngIntEltInertiaDegree(L): RngXPad → RngIntEltRamificationDegree(L, K): FldXPad, FldXPad → RngIntEltRamificationDegree(L): FldXPad → RngIntEltRamificationIndex(L): FldXPad → RngIntEltRamificationIndex(L, K): FldXPad, FldXPad → RngIntEltRamificationDegree(L): RngXPad → RngIntEltRamificationDegree(L, K): RngXPad, RngXPad → RngIntEltRamificationIndex(L): RngXPad → RngIntEltRamificationIndex(L, K): RngXPad, RngXPad → RngIntEltDefiningPolynomial(R): RngXPad → RngUPolEltDefiningPolynomial(R): FldXPad → RngUPolEltAbsoluteDegree(F): FldXPad → RngIntEltAbsoluteInertiaDegree(L): FldXPad → RngIntEltAbsoluteInertiaIndex(L): FldXPad → RngIntEltAbsoluteRamificationDegree(L): FldXPad → RngIntEltAbsoluteRamificationIndex(L): FldXPad → RngIntEltAbsoluteDegree(F): RngXPad → RngIntEltAbsoluteInertiaDegree(L): RngXPad → RngIntEltAbsoluteInertiaIndex(L): RngXPad → RngIntEltAbsoluteRamificationDegree(L): RngXPad → RngIntEltAbsoluteRamificationIndex(L): RngXPad → RngIntEltDiscriminantValuation(L): FldXPad → RngIntEltDiscriminantValuation(L, K): FldXPad, FldXPad → RngIntEltDiscriminantValuation(L, K): RngXPad, RngXPad → RngIntEltDiscriminantValuation(L): RngXPad → RngIntEltRamificationPolygon(L): FldXPad → NwtnPgonRamificationPolygon(L, K): FldXPad, FldXPad → NwtnPgonRamificationPolygon(L, K): RngXPad, RngXPad → NwtnPgonRamificationPolygon(L): RngXPad → NwtnPgonRamificationPolygon(f): RngUPolXPadElt[FldXPad] → NwtnPgonRamificationPolygon(f): RngUPolXPadElt[RngXPad] → NwtnPgon- Example: Invariants of Exact p-adic Fields
- Exact \(p\)-adic Elements
GetExactpAdicsPrintPrecision() → RngIntEltSetExactpAdicsPrintPrecision(k): InftySetExactpAdicsPrintPrecision(k): RngIntEltK ! x: FldXPad, Any → FldXPadEltR ! x: RngXPad, Any → RngXPadEltelt<K | x>: FldXPad, Any → FldXPadEltelt<R | x>: RngXPad, Any → RngXPadEltAbsolutePrecision(x): FldXPadElt → RngIntEltAbsolutePrecision(x): RngXPadElt → RngIntEltRelativePrecision(x): FldXPadElt → RngIntEltRelativePrecision(x): RngXPadElt → RngIntEltValuation(x): FldXPadElt → RngIntEltValuation(x): RngXPadElt → RngIntEltWeakValuation(x): RngXPadElt → RngIntEltWeakValuation(x): FldXPadElt → RngIntEltShiftValuation(x, n): FldXPadElt, RngIntElt → FldXPadEltShiftValuation(x, n): RngXPadElt, RngIntElt → RngXPadEltValuationEq(x, n): FldXPadElt, RngIntElt → BoolEltValuationEq(x, n): FldXPadElt, Infty → BoolEltValuationEq(x, n): RngXPadElt, RngIntElt → BoolEltValuationEq(x, n): RngXPadElt, Infty → BoolEltValuationNe(x, n): FldXPadElt, RngIntElt → BoolEltValuationNe(x, n): FldXPadElt, Infty → BoolEltValuationNe(x, n): RngXPadElt, RngIntElt → BoolEltValuationNe(x, n): RngXPadElt, Infty → BoolEltValuationGe(x, n): FldXPadElt, RngIntElt → BoolEltValuationGe(x, n): FldXPadElt, Infty → BoolEltValuationGe(x, n): RngXPadElt, RngIntElt → BoolEltValuationGe(x, n): RngXPadElt, Infty → BoolEltValuationGt(x, n): FldXPadElt, Infty → BoolEltValuationGt(x, n): FldXPadElt, RngIntElt → BoolEltValuationGt(x, n): RngXPadElt, RngIntElt → BoolEltValuationGt(x, n): RngXPadElt, Infty → BoolEltValuationLe(x, n): FldXPadElt, Infty → BoolEltValuationLe(x, n): FldXPadElt, RngIntElt → BoolEltValuationLe(x, n): RngXPadElt, RngIntElt → BoolEltValuationLe(x, n): RngXPadElt, Infty → BoolEltValuationLt(x, n): FldXPadElt, RngIntElt → BoolEltValuationLt(x, n): FldXPadElt, Infty → BoolEltValuationLt(x, n): RngXPadElt, Infty → BoolEltValuationLt(x, n): RngXPadElt, RngIntElt → BoolEltIsUnit(x): RngXPadElt → BoolEltIsUnit(x): FldXPadElt → BoolEltIsIntegral(x): RngXPadElt → BoolEltIsIntegral(x): FldXPadElt → BoolEltIsWeaklyZero(x): StrAnyXPadElt → BoolEltIsWeaklyEqual(x, y): StrAnyXPadElt, StrAnyXPadElt → BoolEltIsDefinitelyZero(x): StrAnyXPadElt → BoolEltIsDefinitelyEqual(x, y): StrAnyXPadElt, StrAnyXPadElt → BoolEltCoerceAndLift(S, x): StrAnyXPad, Any → StrAnyXPadElt- Arithmetic with Elements
x * y: FldXPadElt, FldXPadElt → FldXPadEltx * y: RngXPadElt, RngXPadElt → RngXPadEltx + y: FldXPadElt, FldXPadElt → FldXPadEltx + y: RngXPadElt, RngXPadElt → RngXPadElt- y: FldXPadElt → FldXPadElt- y: RngXPadElt → RngXPadEltx - y: RngXPadElt, RngXPadElt → RngXPadEltx - y: FldXPadElt, FldXPadElt → FldXPadEltx ^ n: FldXPadElt, RngIntElt → FldXPadEltx ^ n: RngXPadElt, RngIntElt → RngXPadEltx / y: FldXPadElt, FldXPadElt → FldXPadEltx / y: RngXPadElt, RngXPadElt → RngXPadEltx div y: RngXPadElt, RngXPadElt → RngXPadEltQuotrem(x, y): RngXPadElt, RngXPadElt → RngXPadElt, RngXPadEltGCD(x, y): FldXPadElt, FldXPadElt → FldXPadEltGcd(x, y): FldXPadElt, FldXPadElt → FldXPadEltGreatestCommonDivisor(x, y): FldXPadElt, FldXPadElt → FldXPadEltGCD(x, y): RngXPadElt, RngXPadElt → RngXPadEltGcd(x, y): RngXPadElt, RngXPadElt → RngXPadEltGreatestCommonDivisor(x, y): RngXPadElt, RngXPadElt → RngXPadEltExtendedGreatestCommonDivisor(x, y): FldXPadElt, FldXPadElt → FldXPadElt, FldXPadElt, FldXPadEltXGCD(x, y): FldXPadElt, FldXPadElt → FldXPadElt, FldXPadElt, FldXPadEltXgcd(x, y): FldXPadElt, FldXPadElt → FldXPadElt, FldXPadElt, FldXPadEltExtendedGreatestCommonDivisor(x, y): RngXPadElt, RngXPadElt → RngXPadElt, RngXPadElt, RngXPadEltXGCD(x, y): RngXPadElt, RngXPadElt → RngXPadElt, RngXPadElt, RngXPadEltXgcd(x, y): RngXPadElt, RngXPadElt → RngXPadElt, RngXPadElt, RngXPadElt- Example: Arithmetic with Exact p-adic Elements
- Polynomials over Exact \(p\)-adic Rings and Fields
- Exact Polynomial Rings
- Polynomials
R ! f: RngUPolXPad, Any → RngUPolXPadEltBaseRing(f): RngUPolXPadElt → RngCoefficientRing(f): RngUPolXPadElt → RngCanChangeRing(f, R): RngUPolXPadElt, Rng → BoolElt, RngUPolXPadEltChangeRing(f, R): RngUPolXPadElt, Rng → RngUPolXPadEltDegree(f): RngUPolXPadElt → RngIntEltWeakDegree(f): RngUPolXPadElt → RngIntEltCoefficient(f, i): RngUPolXPadElt, RngIntElt → RngEltCoefficients(f): RngUPolXPadElt → SeqEnumExactPolynomial(f): RngUPolXPadElt → RngUPolXPadEltExactPolynomial(f): RngUPolElt → RngUPolXPadEltEvaluate(f, x): RngUPolXPadElt, Any → RngAnyXPadEltDerivative(f, m): RngUPolXPadElt, RngIntElt → RngUPolXPadEltDerivative(f): RngUPolXPadElt → RngUPolXPadEltDiscriminant(f): RngUPolXPadElt → RngAnyXPadEltResultant(f, g): RngUPolXPadElt, RngUPolXPadElt → RngAnyXPadEltIsInertial(f): RngUPolXPadElt → BoolEltIsEisenstein(f): RngUPolXPadElt → BoolEltIsWeaklyZero(f): StrAnyXPadElt → BoolEltIsWeaklyEqual(f, g): StrAnyXPadElt, StrAnyXPadElt → BoolEltIsDefinitelyZero(f): StrAnyXPadElt → BoolEltIsDefinitelyEqual(f, g): StrAnyXPadElt, StrAnyXPadElt → BoolEltCoerceAndLift(S, x): StrAnyXPad, Any → StrAnyXPadElt
- Arithmetic
- Factorization and Roots
NewtonPolygon(f): RngUPolXPadElt[RngXPad] → NwtnPgonNewtonPolygon(f): RngUPolXPadElt[FldXPad] → NwtnPgonRoots(f, R): RngUPolElt, FldXPad → SeqEnumRoots(f, R): RngUPolElt, RngXPad → SeqEnumRoots(f, R): RngUPolXPadElt, RngXPad → SeqEnumRoots(f, R): RngUPolXPadElt, FldXPad → SeqEnumRoots(f): RngUPolXPadElt[FldXPad] → SeqEnumRoots(f): RngUPolXPadElt[RngXPad] → SeqEnumHasRoot(f): RngUPolXPadElt[RngXPad] → BoolElt, RngXPadEltHasRoot(f): RngUPolXPadElt[FldXPad] → BoolElt, FldXPadEltHasRoot(f): RngUPolElt[RngXPad] → BoolElt, RngXPadEltHasRoot(f): RngUPolElt[FldXPad] → BoolElt, FldXPadEltHasRoot(f, R): RngUPolElt, FldXPad → BoolElt, FldXPadEltHasRoot(f, R): RngUPolElt, RngXPad → BoolElt, RngXPadEltHasRoot(f, R): RngUPolXPadElt, FldXPad → BoolElt, FldXPadEltHasRoot(f, R): RngUPolXPadElt, RngXPad → BoolElt, RngXPadEltFactorization(f, R): RngUPolXPadElt, FldXPad → SeqEnum, RngXPadElt, SeqEnumFactorisation(f, R): RngUPolXPadElt, FldXPad → SeqEnum, RngXPadElt, SeqEnumFactorisation(f, R): RngUPolXPadElt, RngXPad → SeqEnum, RngXPadElt, SeqEnumFactorization(f, R): RngUPolXPadElt, RngXPad → SeqEnum, RngXPadElt, SeqEnumFactorization(f, R): RngUPolElt, RngXPad → SeqEnum, RngXPadElt, SeqEnumFactorisation(f, R): RngUPolElt, RngXPad → SeqEnum, RngXPadElt, SeqEnumFactorisation(f, R): RngUPolElt, FldXPad → SeqEnum, RngXPadElt, SeqEnumFactorization(f, R): RngUPolElt, FldXPad → SeqEnum, RngXPadElt, SeqEnumFactorization(f): RngUPolXPadElt[RngXPad] → SeqEnum, RngXPadElt, SeqEnumFactorization(f): RngUPolXPadElt[FldXPad] → SeqEnum, FldXPadElt, SeqEnumFactorisation(f): RngUPolXPadElt[RngXPad] → SeqEnum, RngXPadElt, SeqEnumFactorisation(f): RngUPolXPadElt[FldXPad] → SeqEnum, FldXPadElt, SeqEnumIsIrreducible(f): RngUPolXPadElt[RngXPad] → BoolElt, RecIsIrreducible(f): RngUPolXPadElt[FldXPad] → BoolElt, RecIsIrreducible(f): RngUPolElt[RngXPad] → BoolElt, RecIsIrreducible(f): RngUPolElt[FldXPad] → BoolElt, RecIsHenselLiftable(f, x): RngUPolElt, FldXPadElt → BoolElt, FldXPadEltIsHenselLiftable(f, x): RngUPolElt, RngXPadElt → BoolElt, RngXPadEltIsHenselLiftable(f, x): RngUPolXPadElt, FldXPadElt → BoolElt, FldXPadEltIsHenselLiftable(f, x): RngUPolXPadElt, RngXPadElt → BoolElt, RngXPadEltRamificationResidualPolynomial(f, face): RngUPolElt[FldXPad], NwtnPgonFace → RngUPolEltRamificationResidualPolynomial(f, face): RngUPolElt[RngXPad], NwtnPgonFace → RngUPolEltRamificationResidualPolynomial(f, face): RngUPolXPadElt[FldXPad], NwtnPgonFace → RngUPolEltRamificationResidualPolynomial(f, face): RngUPolXPadElt[RngXPad], NwtnPgonFace → RngUPolEltRamificationResidualPolynomials(f): RngUPolElt[FldXPad] → SeqEnum, NwtnPgonRamificationResidualPolynomials(f): RngUPolElt[RngXPad] → SeqEnum, NwtnPgonRamificationResidualPolynomials(f): RngUPolXPadElt[FldXPad] → SeqEnum, NwtnPgonRamificationResidualPolynomials(f): RngUPolXPadElt[RngXPad] → SeqEnum, NwtnPgon- Example: Factorization over Exact p-adic Fields
- Bibliography