Quotients#

Quotient(I, zbJ): AlgEtQIdl, SeqEnum[AlgEtQElt] -> GrpAb, Map#

Given an ideal \(I\) and the ZBasis of an ideal or order \(J\) such that \(J \subset I\), returns the abelian group \(Q=I/J\) together with the quotient map \(q:I\mapsto Q\).

Quotient(I, J): AlgEtQIdl, AlgEtQIdl -> GrpAb, Map#

Given fractional ideals \(J \subset I\), returns the abelian group \(Q=I/J\) together with the quotient map \(q:I\mapsto Q\).

Quotient(S, zbJ): AlgEtQOrd, SeqEnum[AlgEtQElt] -> GrpAb, Map#

Given an order \(S\) and the ZBasis of an ideal \(J\) such that \(J \subset S\), returns the abelian group \(Q=S/J\) together with the quotient map \(q:S\mapsto Q\). 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 \(q:S \mapsto S/I\) (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) \(S\mapsto F\).

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:

  • \(P\) 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 \(K^d=V\) and the natural surjection \(I\mapsto V\) (with preimages).

Example: Quotients Residues (ex-c0d808)#
> _<x> := PolynomialRing(Integers());
> f := (x^8+16)*(x^8+81);
> A := EtaleAlgebra(f);
> O := MaximalOrder(A);
> // Choose a prime above 2
> P := PrimesAbove(2*O)[1];
> // Residue field
> F, pi := ResidueField(P);
> F;
Finite field of size 2
> // Residue ring
> G, q := ResidueRing(O,P);
> G;
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
2*G.1 = 0

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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 \(K^d=V\) and the natural surjection \(I\mapsto V\) (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 \(K^d=V\) and the natural surjection \(I\mapsto V\) (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 \(K^d=V\) and the natural surjection \(I\mapsto V\) (with preimages).