Ideals of Z
ideal< R | a > : RngInt, RngIntElt -> RngIntRes
Example RngIntRes_residue-ring (H19E1)
Z as a Number Field Order
Decomposition(R, p) : RngInt, RngIntElt -> SeqEnum
Generator(I) : RngInt -> RngIntElt
RamificationIndex(I, p) : RngInt, RngIntElt -> RngIntElt
Degree(I) : RngInt -> RngIntElt
TwoElementNormal(I) : RngInt -> RngIntElt, RngIntElt
ChineseRemainderTheorem(I, J, a, b) : RngInt, RngInt, RngIntElt, RngIntElt -> RngIntElt
Valuation(x, I) : RngIntElt, RngInt -> RngIntElt
ClassRepresentative(I) : RngInt -> RngInt
Creation
quo< Z | I > : RngInt, RngInt -> RngIntRes
quo< Z | m > : RngInt, RngIntElt -> RngIntRes
ResidueClassRing(m) : RngIntElt -> RngIntRes
ResidueClassRing(Q) : RngIntEltFact -> RngIntRes
Example RngIntRes_residue-ring (H19E2)
Coercion
Example RngIntRes_Coercion (H19E3)
Elementary Invariants
Modulus(R) : RngIntRes -> RngInt
FactoredModulus(R) : RngIntRes -> RngIntEltFact
Structure Operations
AdditiveGroup(R) : RngIntRes -> GrpAb, Map
MultiplicativeGroup(R) : RngIntRes -> GrpAb, Map
sub< R | n > : RngIntRes, RngIntResElt -> RngIntRes
Set(R) : RngIntRes -> SetEnum
Homomorphisms
hom< R -> S | > : RngIntRes, Rng -> Map
Elements of Residue Class Rings
Creation
elt< R | k > : RngIntRes, RngIntElt -> RngIntResElt
R ! k : RngIntRes, RngIntElt -> RngIntResElt
Random(R) : RngIntRes -> RngIntResElt
Solving Equations over Z/mZ
Solution(a, b) : RngIntResElt, RngIntResElt -> RngIntResElt
IsSquare(n) : RngIntResElt -> BoolElt, RngIntResElt
Sqrt(a) : RngIntResElt -> RngIntResElt
AllSquareRoots(a) : RngIntResElt -> [ RngIntResElt ]
Example RngIntRes_element-ops (H19E4)
Ideal Operations
ideal< R | a1, ..., ar > : RngIntRes, RngIntResElt, ..., RngIntResElt -> RngIntRes
GreatestCommonDivisor(a, b) : RngIntResElt, RngIntResElt -> RngIntResElt
LeastCommonMultiple(a, b) : RngIntResElt, RngIntResElt -> RngIntResElt
LeastCommonMultiple(Q) : [RngIntResElt] -> RngIntResElt
The Unit Group
UnitGroup(R) : RngIntRes -> GrpAb, Map
IsPrimitive(n) : RngIntResElt -> BoolElt
PrimitiveElement(R) : RngIntRes -> RngIntResElt
Order(a) : RngIntResElt -> RngIntElt
Normalize(x) : RngIntRes -> RngIntResElt, RngIntResElt
Example RngIntRes_unit-group (H19E5)
Example RngIntRes_cyclic-unit-group (H19E6)
Creation
DirichletGroup(N) : RngIntElt -> GrpDrch
DirichletGroup(N,R) : RngIntElt, Rng -> GrpDrch
DirichletGroup(N,R,z,r) : RngIntElt, Rng, RngElt, RngIntElt -> GrpDrch
FullDirichletGroup(N) : RngIntElt -> GrpDrch
BaseExtend(G, R) : GrpDrch, Rng -> GrpDrch
AssignNames(~G, S) : GrpDrch, [MonStgElt] ->
Element Creation
Elements(G) : GrpDrch -> [GrpDrchElt]
Random(G) : GrpDrch -> GrpDrchElt
G . i : GrpDrch, RngIntElt -> GrpDrchElt
G ! x : GrpDrch, Any -> GrpDrchElt
KroneckerCharacter(D) : RngIntElt -> GrpDrchElt
Properties of Dirichlet Groups
BaseRing(G) : GrpDrch -> Rng
Modulus(G) : GrpDrch -> RngIntElt
Order(G) : GrpDrch -> RngIntElt
Exponent(G) : GrpDrch -> RngIntElt
AbelianGroup(G) : GrpDrch -> GrpAb, Map
NumberOfGenerators(G) : GrpDrch -> RngIntElt
Generators(G) : GrpDrch -> [GrpDrchElt]
G . i : GrpDrch, RngIntElt -> GrpDrchElt
UnitGenerators(G) : GrpDrch -> [RngIntElt]
Properties of Elements
BaseRing(chi) : GrpDrchElt -> Rng
Modulus(chi) : GrpDrchElt -> RngIntElt
Conductor(chi) : GrpDrchElt -> RngIntElt
ElementToSequence(chi) : GrpDrchElt -> SeqEnum
x eq y : GrpDrchElt, GrpDrchElt -> BoolElt
Order(chi) : GrpDrchElt -> RngIntElt
IsTrivial(chi) : GrpDrchElt -> BoolElt
IsPrimitive(chi) : GrpDrchElt -> BoolElt
AssociatedPrimitiveCharacter(chi) : GrpDrchElt -> GrpDrchElt
IsEven(chi) : GrpDrchElt -> BoolElt
IsOdd(chi) : GrpDrchElt -> BoolElt
IsTotallyEven(chi) : GrpDrchElt -> BoolElt
Decomposition(chi) : GrpDrchElt -> List
GaloisConjugacyRepresentatives(G) : GrpDrch -> [GrpDrchElt]
MinimalBaseRingCharacter(chi) : GrpDrchElt -> GrpDrchElt
Evaluation
Evaluate(chi,n) : GrpDrchElt, RngIntElt -> RngElt
ValueList(chi) : GrpDrchElt -> [RngElt]
ValuesOnUnitGenerators(chi) : GrpDrchElt -> [RngElt]
OrderOfRootOfUnity(r, n) : RngElt, RngIntElt -> RngIntElt
Arithmetic
x * y : GrpDrchElt, GrpDrchElt -> GrpDrchElt
x ^ n : GrpDrchElt, RngIntElt -> GrpDrchElt
x ^ phi : GrpDrchElt, Map -> GrpDrchElt
Sqrt(x) : GrpDrchElt -> GrpDrchElt
Example
Example RngIntRes_Dirichlet (H19E7)
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013