Quotients

Quotient(I, zbJ) : AlgEtQIdl, SeqEnum[AlgEtQElt] -> GrpAb, Map
Given an ideal I and the ZBasis of an ideal or order J such that J ⊂I, returns the abelian group Q=I/J together with the quotient map q:I |-> Q.
Quotient(I, J) : AlgEtQIdl, AlgEtQIdl -> GrpAb, Map
Given fractional ideals J ⊂I, returns the abelian group Q=I/J together with the quotient map q:I |-> Q.
Quotient(S, zbJ) : AlgEtQOrd, SeqEnum[AlgEtQElt] -> GrpAb, Map
Given an order S and the ZBasis of an ideal J such that J ⊂S, returns the abelian group Q=S/J together with the quotient map q:S |-> Q. The ideal J can also be an order.
ResidueRing(S, I) : AlgEtQOrd, AlgEtQIdl -> GrpAb , Map
Given an integral ideal I of S, returns the abelian group S/I and the quotient map q:S |-> S/I (with preimages). Important: the domain of q is the Algebra of S, since the elements of S are expressed as elements of A. We stress that the output is a group and does not have a multiplication. This can be obtained by first taking preimages, doing the multiplication, and then applying the projection.
ResidueField(P) : AlgEtQIdl -> FldFin, Map
Given P a prime of S, returns a finite field F isomorphic to S/P and a surjection (with inverse) S |-> F.
PrimitiveElementResidueField(P) : AlgEtQIdl->AlgEtQElt
Returns an element of the ideal P that maps to the primitive element of the residue field S/P, that is a multiplicative generator of (S/P) * .
QuotientVS(I, J, P) : AlgEtQOrd, AlgEtQOrd, AlgEtQIdl -> ModRng, Map
Let I, J be orders, P a fractional R-ideal such that:
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P is prime of some order R with residue field K;
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J in I and I/J is a vector space V over K, say of dimension d.

The function returns the KModule Kd=V and the natural surjection I |-> V (with preimages).

QuotientVS(I, J, P) : AlgEtQOrd, AlgEtQIdl, AlgEtQIdl -> ModRng, Map
Let I be an order, J and P be fractional R-ideals such that:
-
P is prime of some order R, with residue field K;
-
J in I and I/J is a vector space V over K, say of dimension d.

The function returns the KModule Kd=V and the natural surjection I |-> V (with preimages).

QuotientVS(I, J, P) : AlgEtQIdl, AlgEtQOrd, AlgEtQIdl -> ModRng, Map
Let J be an order, I and P be fractional R-ideals such that:
-
P is prime of some order R, with residue field K;
-
J in I and I/J is a vector space V over K, say of dimension d.

The function returns the KModule Kd=V and the natural surjection I |-> V (with preimages).

QuotientVS(I, J, P) : AlgEtQIdl, AlgEtQIdl, AlgEtQIdl -> ModRng, Map
Let I, J, P be fractional R-ideals such that:
-
P is prime of some order R;
-
J in I and I/J is a vector space over R/P, say of dimension d;

The function returns the KModule Kd=V and the natural surjection I |-> V (with preimages).

V2.29, 21 October 2025