Creation of Associative Orders
Order(R, S) : Rng, SeqEnum[AlgAssVElt] -> AlgAssVOrd
Order(S) : SeqEnum[AlgAssVElt[FldAlg]] -> AlgAssVOrd
Order(S, I) : SeqEnum[AlgAssVElt[FldAlg]], SeqEnum[RngOrdFracIdl] -> AlgAssVOrd
Order(A, m, I) : AlgAssV[FldOrd], AlgMatElt[FldOrd], SeqEnum[RngOrdFracIdl] -> AlgAssVOrd
Order(A, pm) : AlgAssV[FldOrd], PMat -> AlgAssVOrd
Example AlgAssVOrd_ord_creat_cyc (H73E1)
Example AlgAssVOrd_ord_creat_cyc (H73E2)
MaximalOrder(A) : AlgAssV[FldRat] -> AlgAssVOrd
Example AlgAssVOrd_max_ord (H73E3)
Attributes of Associative Orders
BaseRing(O) : AlgAssVOrd -> Rng
Algebra(O) : AlgAssVOrd -> AlgAssV
Degree(O) : AlgAssVOrd -> RngIntElt
Discriminant(O) : AlgAssVOrd[RngOrd] -> RngOrdIdl
FactoredDiscriminant(O) : AlgAssVOrd[RngOrd] -> [Tup]
MultiplicationTable(O) : AlgAssVOrd -> SeqEnum
Module(O) : AlgAssVOrd[RngOrd] -> PMat
TraceZeroSubspace(O) : AlgAssVOrd -> [Tup]
Bases of Associative Orders
Basis(O) : AlgAssVOrd -> SeqEnum
PseudoBasis(O) : AlgAssVOrd[RngOrd] -> SeqEnum
PseudoMatrix(O) : AlgAssVOrd[RngOrd]> -> PMat
ZBasis(O) : AlgAssVOrd[RngOrd] -> [AlgAssVElt]
Example AlgAssVOrd_bases (H73E4)
Predicates on Orders
O1 eq O2 : AlgAssVOrd, AlgAssVOrd -> BoolElt
x in O : AlgAssVElt, AlgAssVOrd -> BoolElt
Operations with Orders
Adjoin(O, x) : AlgAssVOrd, AlgAssVElt -> AlgAssVOrd
O1 + O2 : AlgAssVOrd[RngOrd], AlgAssVOrd[RngOrd] -> AlgAssVOrd
Example AlgAssVOrd_sumandadjoin (H73E5)
Elements of Associative Orders
Creation of Elements
O ! 0 : AlgAssVOrd, RngIntElt -> AlgAssVOrdElt
O ! 1 : AlgAssVOrd, RngIntElt -> AlgAssVOrdElt
O . i : AlgAssVOrd, RngIntElt -> AlgAssVElt
O ! x : AlgAssVOrd, Any -> AlgAssVOrdElt
Random(O) : AlgAssVOrd -> AlgAssVOrdElt
Arithmetic of Elements
x + y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt
x - y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt
- x : AlgAssVOrdElt -> AlgAssVOrdElt
x * y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt
u * c : AlgAssVOrdElt, RngElt -> AlgAssVOrdElt
x / y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVElt
x div y : AlgAssVOrdElt, AlgAssVOrdElt -> AlgAssVOrdElt
x ^ n : AlgAssVOrdElt, RngIntElt -> AlgAssVOrdElt
Predicates on Elements
x eq y : AlgAssVOrdElt, AlgAssVOrdElt -> BoolElt
x ne y : AlgAssVOrdElt, AlgAssVOrdElt -> BoolElt
IsZero(x) : AlgAssVOrdElt -> BoolElt
IsUnit(a) : AlgAssVOrdElt -> BoolElt
IsScalar(x) : AlgAssVOrdElt -> BoolElt, RngElt
Other Operations with Elements
ElementToSequence(x) : AlgAssVOrdElt -> SeqEnum
Norm(x) : AlgAssVOrdElt -> RngElt
Trace(x) : AlgAssVOrdElt -> RngElt
LeftRepresentationMatrix(e) : AlgAssVOrdElt -> AlgMatElt
RepresentationMatrix(a) : AlgAssVOrdElt -> AlgMatElt
CharacteristicPolynomial(x) : AlgAssVOrdElt -> RngUPolElt
MinimalPolynomial(x) : AlgAssVOrdElt -> RngUPolElt
Creation of Ideals
lideal<O | E> : AlgAssVOrd, [AlgAssVOrdElt] -> AlgAssVOrdIdl
O * e : AlgAssVOrd, RngElt -> AlgAssVOrdIdl
RandomRightIdeal(O) : AlgAssVOrd -> AlgAssVOrdIdl
Attributes of Associative Ideals
Algebra(I) : AlgAssVOrdIdl -> AlgAssV
Order(I) : AlgAssVOrdIdl -> AlgAssVOrd
LeftOrder(I) : AlgAssVOrdIdl[RngOrd] -> AlgAssVOrd
Basis(I) : AlgAssVOrdIdl -> SeqEnum
BasisMatrix(I) : AlgAssVOrdIdl -> AlgMatElt
PseudoBasis(I) : AlgAssVOrdIdl[RngOrd] -> SeqEnum
PseudoMatrix(I) : AlgAssVOrdIdl[RngOrd] -> PMat
ZBasis(I) : AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]
Denominator(I) : AlgAssVOrdIdl -> RngElt
Arithmetic of Ideals
I + J : AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl
I * J: AlgAssVOrdIdl, AlgAssVOrdIdl -> AlgAssVOrdIdl, AlgAssVOrdIdl
a * I: RngElt, AlgAssVOrdIdl -> AlgAssVOrdIdl
Colon(J, I): AlgAssVOrdIdl[RngOrd], AlgAssVOrdIdl[RngOrd] -> PMat
MultiplicatorRing(I): AlgAssVOrdIdl -> AlgAssVOrd
Predicates on Ideals
IsLeftIdeal(I) : AlgAssVOrdIdl -> BoolElt
I eq J : AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt
I subset J : AlgAssVOrdIdl, AlgAssVOrdIdl -> BoolElt
a in I : AlgAssVElt, AlgAssVOrdIdl -> BoolElt
Other Operations on Ideals
Norm(I) : AlgAssVOrdIdl[RngOrd] -> RngOrdIdl