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 (H61E1)
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
Example RngLoc_el_creation_unram (H61E2)
Creation Functions for Totally Ramified Extensions
TotallyRamifiedExtension(L, f) : RngPad, RngUPolElt -> RngPad
IsEisenstein(f) : RngUPolElt -> BoolElt
Example RngLoc_el_creation_ram (H61E3)
Creation Functions for Unbounded Precision Extensions
ext<L | m> : RngPad, Map -> RngPad
Example RngLoc_el_creation_map (H61E4)
Miscellaneous Creation Functions
IntegerRing(F) : FldPad -> RngPad
FieldOfFractions(R) : RngPad -> FldPad
Attributes of Local Rings and Fields
L`DefaultPrecision : RngLoc -> RngIntElt
Elementary Invariants
Prime(L) : RngPad -> RngIntElt
InertiaDegree(L) : RngPad -> RngIntElt
InertiaDegree(K, L) : RngPad, RngPad -> RngIntElt
RamificationDegree(L) : RngPad -> RngIntElt
RamificationDegree(K, L) : RngPad, 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
UniformizingElement(L) : RngPad -> RngPadElt
L . 1 : RngPad -> RngPadElt
Precision(L) : RngPad -> RngIntElt
Example RngLoc_elinvar (H61E5)
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, RngIntElt -> RngPad
L eq K : RngPad, RngPad -> BoolElt
L ne K : RngPad, RngPad -> BoolElt
Example RngLoc_strop (H61E6)
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 (H61E7)
Example RngLoc_eltcons_seq_weird (H61E8)
Element Decomposers
ElementToSequence(x) : RngPadElt -> [ RngElt ]
Coefficient(x, i) : RngPadElt, RngIntElt -> RngPadElt
Example RngLoc_gal-desc (H61E9)
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 / y : RngPadElt, RngPadElt -> RngPadElt
IsExactlyDivisible(x, y) : RngPadElt, RngPadElt -> BoolElt, RngPadElt
Example RngLoc_Division (H61E10)
Equality and Membership
x eq y : RngPadResElt, RngPadResElt -> BoolElt
x ne y : RngPadResElt, RngPadResElt -> BoolElt
x in L : ., RngLoc -> BoolElt
x notin L : ., RngLoc -> BoolElt
Example RngLoc_unram-ext (H61E11)
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) : RngPadElt, RngIntElt -> RngPadElt
Expand(x) : RngPadElt -> RngPadElt
Valuation(x) : RngPadElt -> RngIntElt
Example RngLoc_ofe (H61E12)
Logarithms and Exponentials
Log(x) : RngPadElt -> RngPadElt
Exp(x) : RngPadElt -> RngPadElt
Example RngLoc_log (H61E13)
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
GaloisImage(x, i) : RngPadElt, RngIntElt -> RngPadElt
Example RngLoc_agm (H61E14)
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 (H61E15)
Hensel Lifting of Roots
NewtonPolygon(f) : RngUPolElt -> NwtnPgon
ValuationsOfRoots(f) : RngUPolElt -> SeqEnum[<FldRatElt, RngIntElt>]
Example RngLoc_newton-polygon (H61E16)
HenselLift(f, x) : RngUPolElt, RngPadElt -> RngPadElt
Example RngLoc_Hensel (H61E17)
Functions returning Roots
Roots(f) : RngUPolElt -> [ <RngLocElt, RngIntElt> ]
HasRoot(f) : RngUPolElt -> BoolElt, RngLocElt
Example RngLoc_ramified-ext (H61E18)
Factorization
HenselLift(f, s) : RngUPolElt, [RngUPolElt] -> [RngUPolElt]
Example RngLoc_Poly-Hensel (H61E19)
IsIrreducible(f) : RngUPolElt -> BoolElt
SquareFreeFactorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
Factorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt > ]
SuggestedPrecision(f) : RngUPolElt -> RngIntElt
Example RngLoc_factors-precision (H61E20)
Example RngLoc_Factors (H61E21)
Automorphisms of Local Rings and Fields
Automorphisms(L) : RngLoc -> [Map]
AutomorphismGroup(L) : RngLoc -> GrpPerm, Map
Example RngLoc_units-autos (H61E22)
Completions
Completion(O, P) : RngOrd, RngOrdIdl -> RngLoc, Map
LocalRing(P, k) : RngOrdIdl, RngIntElt -> RngLoc, Map
Example RngLoc_completion (H61E23)