Local Rings and Fields with Infinite Precision
Creation of Local Rings and Fields
Elementary Creation Functions
pAdicRing(p, n) : RngIntElt, RngIntElt -> RngLoc
UnramifiedExtension(L, f) : RngLoc, RngIntElt -> RngLoc
TotallyRamifiedExtension(L, g) : RngLoc, RngUPolElt -> RngLoc
ext<L | f, g> : RngLoc, RngIntElt, RngUPolElt -> RngLoc
IsEisenstein(g) : RngUPolElt -> BoolElt
IsInertial(g) : RngUPolElt -> BoolElt
Example RngLoc_el_creation (H55E1)
Between Ring and Field
IntegerRing(L) : FldLoc -> RngLoc
FieldOfFractions(L) : RngLoc -> FldLoc
Attributes of Local Rings and Fields
L`DefaultPrecision
L`SeriesPrinting
Example RngLoc_series-print (H55E2)
More Creation Functions
LocalRing(p, f, n) : RngIntElt, RngIntElt, RngIntElt -> RngLoc
LocalRing(L, f, n) : RngLoc, RngIntElt -> RngLoc
LocalRing(p, g, n) : RngIntElt, RngUPolElt, RngIntElt -> RngLoc
LocalRing(L, g, n) : RngLoc, RngUPolElt, RngIntElt -> RngLoc
LocalRing(p, f, g, n) : RngIntElt, RngIntElt, RngUPolElt, RngIntElt -> RngLoc
Example RngLoc_creation-ex (H55E3)
Elementary Invariants
Prime(L) : RngLoc -> RngIntElt
InertiaDegree(L) : RngLoc -> RngIntElt
RamificationDegree(L) : RngLoc -> RngIntElt
TotalDegree(L) : RngLoc -> RngIntElt
Degree(L) : RngLoc -> RngIntElt
EisensteinPolynomial(L) : RngLoc -> RngUPolElt
InertialPolynomial(L) : RngLoc -> RngUPolElt
DefiningPolynomial(L) : RngLoc -> RngUPolElt
InertiaRing(L) : RngLoc -> RngLoc
InertiaField(L) : FldLoc -> FldLoc
PrimeRing(L) : RngLoc -> RngLoc
ResidueClassField(L) : RngLoc -> FldFin, Map
UniformizingElement(L) : RngLoc -> RngLocElt
InertiaElement(L) : RngLoc -> RngLocElt
Precision(L) : RngLoc -> RngIntElt
Example RngLoc_elinvar (H55E4)
Operations on Structures
AssignNames(~L, S) : RngLoc, SeqEnum ->
Characteristic(L) : RngLoc -> RngIntElt
# L : RngLoc -> RngIntElt
Name(L, k) : RngLoc, RngIntElt -> RngLocElt
ChangePrecision(L, r) : RngLoc, RngIntElt -> RngLoc
L eq K : RngLoc, RngLoc -> BoolElt
L ne K : RngLoc, RngLoc -> BoolElt
Example RngLoc_strop (H55E5)
Element Constructions and Conversions
Constructions
Zero(L) : RngLoc -> RngLocElt
One(L) : RngLoc -> RngLocElt
Random(L) : RngLoc -> RngLocElt
Representative(L) : RngLoc -> RngLocElt
elt<L | u> : RngLoc, RngElt -> RngLocElt
elt<L | s> : RngLoc, [ RngElt ] -> RngLocElt
elt<L | u, r> : RngLoc, RngElt, RngIntElt -> RngLocElt
elt<L | v, u> : RngLoc, RngIntElt, RngElt -> RngLocElt
elt<L | v, u, r> : RngLoc, RngIntElt, RngElt, RngIntElt -> RngLocElt
BigO(x) : RngLocElt -> RngLocElt
Example RngLoc_eltcons (H55E6)
Sequence Conversions
LocseqInert(x) : RngLoc -> [ RngLocElt ]
InertseqpAdic(x) : RngLoc -> [ RngLocElt ]
Locseq(x) : RngLoc -> [ [ RngLocElt ] ]
ElementToSequence(x) : RngLocElt -> [ RngElt ]
Coefficients(x) : RngLocElt -> [ RngLocElt ]
Coefficient(x, i) : RngLocElt, RngIntElt -> RngLocElt
Example RngLoc_gal-desc (H55E7)
Arithmetic
- x : RngLocElt -> RngLocElt
x + y : RngLocElt, RngLocElt -> RngLocElt
x - y : RngLocElt, RngLocElt -> RngLocElt
x * y : RngLocElt, RngLocElt -> RngLocElt
x ^ k : RngLocElt, RngIntElt -> RngLocElt
x div y : RngLocElt, RngLocElt -> RngLocElt
Example RngLoc_Division (H55E8)
Equality and Membership
x eq y : RngLocElt, RngLocElt -> BoolElt
x ne y : RngLocElt, RngLocElt -> BoolElt
x in L : ., RngLoc -> BoolElt
x notin L : ., RngLoc -> BoolElt
Example RngLoc_unram-ext (H55E9)
Properties
IsZero(x) : RngLocElt -> BoolElt
IsOne(x) : RngLocElt -> BoolElt
IsMinusOne(x) : RngLocElt -> BoolElt
IsUnit(x) : RngLocElt -> BoolElt
IsIntegral(x) : RngLocElt -> BoolElt
Precision and Valuation
Parent(x) : RngLocElt -> RngLoc
Precision(x) : RngLocElt -> RngIntElt
AbsolutePrecision(x) : RngLocElt -> RngIntElt
RelativePrecision(x) : RngLocElt -> RngIntElt
ChangePrecision(x, m) : RngLocElt, RngIntElt -> RngLocElt
Expand(x) : RngLocElt -> RngLocElt
Valuation(x) : RngLocElt -> RngIntElt
Denominator(x) : RngLocElt -> RngIntElt
Example RngLoc_ofe (H55E10)
Logarithms and Exponentials
Log(x) : RngLocElt -> RngLocElt
Exp(x) : RngLocElt -> RngLocElt
Example RngLoc_log (H55E11)
Norm and Trace Functions
Norm(x) : RngLocElt -> RngLocElt
Trace(x) : RngLocElt -> RngLocElt
MinimalPolynomial(x) : RngLocElt -> RngUPolElt
GaloisImage(x, i) : RngLocElt, RngIntElt -> RngLocElt
Linear Algebra
ManhattanForm(M) : Mtrx -> Mtrx, AlgMatElt
Example RngLoc_ech-man (H55E12)
Roots of Elements
HasPRoot(L) : RngLoc -> BoolElt
SquareRoot(x) : RngLocElt -> RngLocElt
IsSquare(x) : RngLocElt -> BoolElt
Root(x, n) : RngLocElt, RngIntElt -> RngLocElt
IsPower(x, n) : RngLocElt, RngIntElt -> RngLocElt
Operations for Polynomials
GreatestCommonDivisor(g, h) : RngUPolElt, RngUPolElt -> RngUPolElt
Example RngLoc_gcd (H55E13)
Hensel Lifting of Roots
NewtonPolygon(g) : RngUPolElt -> NwtnPgon
ValuationsOfRoots(g) : RngUPolElt -> SeqEnum[<FldRatElt, RngIntElt>]
Example RngLoc_newton-polygon (H55E14)
HenselLift(g, r) : RngUPolElt, RngLocElt -> RngLocElt
Example RngLoc_Hensel (H55E15)
Functions returning roots
Roots(g) : RngUPolElt -> [ <RngLocElt, RngIntElt> ]
HasRoot(g) : RngUPolElt -> BoolElt, RngLocElt
Example RngLoc_ramified-ext (H55E16)
Factorization
HenselLift(f, g, h) : RngUPolElt, RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt
Example RngLoc_Poly-Hensel (H55E17)
IsIrreducible(g) : RngUPolElt -> BoolElt
SquareFreeFactorization(g) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
Factorization(g) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
SuggestedPrecision(f) : RngUPolElt -> RngIntElt
Example RngLoc_factors-infinite (H55E18)
Example RngLoc_factors-precision (H55E19)
Example RngLoc_Factors (H55E20)
Automorphisms of Local Rings and Fields
Automorphisms(L) : RngLoc -> [Map]
AutomorphismGroup(L) : RngLoc -> GrpPerm, Map
Example RngLoc_units-autos (H55E21)
Completions
Completion(P, n) : RngOrdIdl, RngIntElt -> RngLoc, Map