Overview of the p-adics in Magma
Free Precision Rings and Fields
Creation of Local Rings and Fields
Creation Functions for the p-adics
pAdicRing(p, k) : RngIntElt, RngIntElt -> RngPad
pAdicRing(p) : RngIntElt -> RngPad
pAdicQuotientRing(p, k) : RngIntElt, RngIntElt -> RngPadRes
quo<L | x> : RngPad, RngPadElt -> .
Example RngLoc_el_creation_padic (H47E1)
Creation Functions for Unramified Extensions
UnramifiedExtension(L, n) : RngPad, RngIntElt -> RngPad
UnramifiedQuotientRing(K, k) : FldFin, RngIntElt -> Rng
UnramifiedExtension(L, f) : RngPad, RngUPolElt -> RngPad
IsInertial(f) : RngUPolElt -> BoolElt
HasGNB(R, n, t) : RngPad, RngIntElt, RngIntElt -> BoolElt
CyclotomicUnramifiedExtension(R, f) : FldPad, RngIntElt -> FldPad
Example RngLoc_el_creation_unram (H47E2)
Creation Functions for Totally Ramified Extensions
TotallyRamifiedExtension(L, f) : RngPad, RngUPolElt -> RngPad
IsEisenstein(f) : RngUPolElt -> BoolElt
Example RngLoc_el_creation_ram (H47E3)
Creation Functions for Unbounded Precision Extensions
ext<L | m> : RngPad, Map -> RngPad
Example RngLoc_el_creation_map (H47E4)
Miscellaneous Creation Functions
IntegerRing(F) : FldPad -> RngPad
RingOfIntegers(R) : RngPad -> RngPad
FieldOfFractions(R) : RngPad -> FldPad
SplittingField(f, R) : RngUPolElt[RngInt], RngPad -> RngPad
AbsoluteTotallyRamifiedExtension(R) : RngPad -> RngPad, Map
Other Elementary Constructions
Composite(R, S) : RngPad, RngPad -> RngPad
Attributes of Local Rings and Fields
L`DefaultPrecision : RngPad -> RngIntElt
Elementary Invariants
Prime(L) : RngPad -> RngIntElt
InertiaDegree(L) : RngPad -> RngIntElt
InertiaDegree(K, L) : RngPad, RngPad -> RngIntElt
AbsoluteInertiaDegree(L) : RngPad -> RngIntElt
RamificationDegree(L) : RngPad -> RngIntElt
RamificationDegree(K, L) : RngPad, RngPad -> RngIntElt
AbsoluteRamificationDegree(L) : RngPad -> RngIntElt
AbsoluteDegree(L) : RngPad -> RngIntElt
Degree(L) : RngPad -> RngIntElt
Degree(K, L) : RngPad, RngPad -> RngIntElt
DefiningPolynomial(L) : RngPad -> RngUPolElt
DefiningMap(L) : RngPad -> Map
HasDefiningMap(L) : RngPad -> BoolElt, Map
PrimeRing(L) : RngPad -> RngPad
BaseRing(L) : RngPad -> RngPad
ResidueClassField(L) : RngPad -> FldFin, Map
ResidueSystem(R) : RngPad -> [RngPadElt]
UniformizingElement(L) : RngPad -> RngPadElt
L . 1 : RngPad -> RngPadElt
Precision(L) : RngPad -> RngIntElt
HasPRoot(R) : RngPad -> BoolElt
HasRootOfUnity(L, n) : RngPad, RngIntElt -> BoolElt
Discriminant(R) : RngPad -> RngPadElt
Discriminant(K, k) : RngPad, RngPad -> RngPadElt
AdditiveGroup(R) : RngPadRes -> GrpAb, Map
Example RngLoc_elinvar (H47E5)
Operations on Structures
AssignNames(~L, S) : RngPad, SeqEnum ->
Characteristic(L) : RngPad -> RngIntElt
# L : RngPad -> RngIntElt
Name(L, k) : RngPad, RngIntElt -> RngPadElt
ChangePrecision(L, k) : RngPad, Any -> RngPad
L eq K : RngPad, RngPad -> BoolElt
L ne K : RngPad, RngPad -> BoolElt
Example RngLoc_strop (H47E6)
Ramification Predicates
IsRamified(R) : RngPad -> BoolElt
IsTamelyRamified(R) : RngPad -> BoolElt
Element Constructions and Conversions
Constructions
Zero(L) : RngPad -> RngPadElt
One(L) : RngPad -> RngPadElt
Random(L) : RngPad -> RngPadElt
Representative(L) : RngPad -> RngPadElt
elt<L | u> : RngPad, RngElt -> RngPadElt
elt<L | u, r> : RngPad, RngElt, RngIntElt -> RngPadElt
elt<L | v, u, r> : RngPad, RngIntElt, RngElt, RngIntElt -> RngPadElt
BigO(x) : RngPadElt -> RngPadElt
UniformizingElement(L) : RngPad -> RngPadElt
Example RngLoc_eltcons (H47E7)
Example RngLoc_eltcons_seq_weird (H47E8)
Element Decomposers
ElementToSequence(x) : RngPadElt -> [ RngElt ]
Coefficient(x, i) : RngPadElt, RngIntElt -> RngPadElt
Example RngLoc_gal-desc (H47E9)
Arithmetic
- x : RngPadElt -> RngPadElt
x + y : RngPadElt, RngPadElt -> RngPadElt
x - y : RngPadElt, RngPadElt -> RngPadElt
x * y : RngPadElt, RngPadElt -> RngPadElt
x ^ k : RngPadElt, RngIntElt -> RngPadElt
x div y : RngPadElt, RngPadElt -> RngPadElt
x div:= y : RngPadElt, RngPadElt -> RngPadElt
x / y : RngPadElt, RngPadElt -> RngPadElt
IsExactlyDivisible(x, y) : RngPadElt, RngPadElt -> BoolElt, RngPadElt
Example RngLoc_Division (H47E10)
Equality and Membership
x eq y : RngPadResElt, RngPadResElt -> BoolElt
x ne y : RngPadResElt, RngPadResElt -> BoolElt
x in L : ., RngPad -> BoolElt
x notin L : ., RngPad -> BoolElt
Example RngLoc_unram-ext (H47E11)
Properties
IsZero(x) : RngPadElt -> BoolElt
IsOne(x) : RngPadElt -> BoolElt
IsMinusOne(x) : RngPadElt -> BoolElt
IsUnit(x) : RngPadElt -> BoolElt
IsIntegral(x) : RngPadElt -> BoolElt
Precision and Valuation
Parent(x) : RngPadElt -> RngPad
Precision(x) : RngPadElt -> RngIntElt
AbsolutePrecision(x) : RngPadElt -> RngIntElt
RelativePrecision(x) : RngPadElt -> RngIntElt
ChangePrecision(x, k) : RngUPolElt, RngIntElt -> RngPadElt
Expand(x) : RngPadElt -> RngPadElt
Valuation(x) : RngPadElt -> RngIntElt
Example RngLoc_ofe (H47E12)
Logarithms and Exponentials
Log(x) : RngPadElt -> RngPadElt
Exp(x) : RngPadElt -> RngPadElt
Example RngLoc_log (H47E13)
Norm and Trace Functions
Norm(x) : RngPadElt -> RngPadElt
Norm(x, R) : RngPadElt, RngPad -> RngPadElt
Trace(x) : RngPadElt -> RngPadElt
Trace(x, R) : RngPadElt, RngPad -> RngPadElt
MinimalPolynomial(x) : RngPadElt -> RngUPolElt
MinimalPolynomial(x, R) : RngPadElt, RngPad -> RngUPolElt
CharacteristicPolynomial(x) : RngPadElt -> RngUPolElt
CharacteristicPolynomial(x, R) : RngPadElt, RngPad -> RngUPolElt
GaloisImage(x, i) : RngPadElt, RngIntElt -> RngPadElt
Example RngLoc_agm (H47E14)
Teichmüller Lifts
TeichmuellerLift(u, R) : FldFinElt, RngPadResExt -> RngPadResExtElt
Roots of Elements
SquareRoot(x) : RngPadElt -> RngPadElt
IsSquare(x) : RngPadElt -> BoolElt, RngPadElt
InverseSquareRoot(x) : RngPadElt -> RngPadElt
InverseSquareRoot(x, y) : RngPadElt, RngPadElt -> RngPadElt
Root(x, n) : RngPadElt, RngIntElt -> RngPadElt
IsPower(x, n) : RngPadElt, RngIntElt -> BoolElt, RngPadElt
InverseRoot(x, n) : RngPadElt, RngIntElt -> RngPadElt
InverseRoot(x, y, n) : RngPadElt, RngPadElt, RngIntElt -> RngPadElt
Operations for Polynomials
GreatestCommonDivisor(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
Example RngLoc_gcd (H47E15)
ShiftValuation(f, n) : RngUPolElt, RngIntElt -> RngUPolElt
Hensel Lifting of Roots
NewtonPolygon(f) : RngUPolElt -> NwtnPgon
ValuationsOfRoots(f) : RngUPolElt -> SeqEnum[<FldRatElt, RngIntElt>]
Example RngLoc_newton-polygon (H47E16)
HenselLift(f, x) : RngUPolElt, RngPadElt -> RngPadElt
Example RngLoc_Hensel (H47E17)
Functions returning Roots
Roots(f) : RngUPolElt -> [ <RngPadElt, RngIntElt> ]
HasRoot(f) : RngUPolElt -> BoolElt, RngPadElt
Example RngLoc_ramified-ext (H47E18)
Factorization
HenselLift(f, s) : RngUPolElt, [RngUPolElt] -> [RngUPolElt]
Example RngLoc_Poly-Hensel (H47E19)
IsIrreducible(f) : RngUPolElt -> BoolElt
SquareFreeFactorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
Factorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
SuggestedPrecision(f) : RngUPolElt -> RngIntElt
IsIsomorphic(f, g) : RngUPolElt, RngUPolElt -> BoolElt
Distance(f, g) : RngUPolElt, RngUPolElt -> RngIntElt
Example RngLoc_factors-precision (H47E20)
Example RngLoc_Factors (H47E21)
Automorphisms of Local Rings and Fields
Automorphisms(L) : RngPad -> [Map]
Automorphisms(K, k) : FldPad, FldPad -> [Map]
AutomorphismGroup(L) : RngPad -> GrpPerm, Map
AutomorphismGroup(K, k) : RngPad, RngPad -> GrpPerm, Map
IsNormal(K) : RngPad -> BoolElt
IsNormal(K, k) : RngPad, RngPad -> BoolElt
IsAbelian(K, k) : FldPad, FldPad -> BoolElt
Continuations(m, L) : Map, RngPad -> [Map]
IsIsomorphic(E, K) : RngPad, RngPad -> BooElt
Example RngLoc_units-autos (H47E22)
Completions
Completion(O, P) : RngOrd, RngOrdIdl -> RngPad, Map
LocalRing(P, k) : RngOrdIdl, RngIntElt -> RngPad, Map
Example RngLoc_completion (H47E23)
Unit Group
PrincipalUnitGroupGenerators(R) : RngPad -> SeqEnum
PrincipalUnitGroup(R) : RngPad -> GrpAb, Map
UnitGroup(R) : RngPad -> GrpAb, Map
UnitGroup(F) : FldPad -> GrpAb, Map
UnitGroupGenerators(R) : RngPad -> SeqEnum
UnitGroupGenerators(F) : FldPad -> SeqEnum
pSelmerGroup(p,F) : RngIntElt, FldPad -> GrpAb, Map
Norm Group
NormGroup(R, m) : FldPad, Map -> GrpAb, Map
NormEquation(R, m, b) : FldPad, Map, RngElt -> BoolElt, RngElt
NormEquation(m1, m2, G) : Map, Map, GrpAb -> GrpAb, Map
Norm(m1, m2, G) : Map, Map, GrpAb -> GrpAb
NormKernel(m1, m2) : Map, Map -> GrpAb
Class Fields
ClassField(m, G) : Map, GrpAb -> FldAb
NormGroupDiscriminant(m, G) : Map, GrpAb -> RngIntElt
Extensions
AllExtensions(R, n) : RngPad, RngIntElt -> [RngPad]
NumberOfExtensions(R, n) : RngPad, RngIntElt -> RngIntElt
OreConditions(R, n, j) : RngPad, RngIntElt, RngIntElt -> BoolElt
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012