Chinese Remainder Theorem#
Let \(I\) and \(J\) be integral fractional ideals over the same order \(S\) in an étale algebra. Assume that \(I\) and \(J\) are coprime, that is, \(I+J=S\). Then \(I \cap J = I\cdot J\) and we have a canonical \(S\)-linear isomorphism
- ChineseRemainderTheorem(Is, as): SeqEnum[AlgEtQIdl], SeqEnum[AlgEtQElt] -> AlgEtQElt#
Given a sequence \(Is\) of ideals of \(S\), pairwise coprime, and a sequence \(as\) of elements of \(S\), it returns an element \(e\) such that \(e-as[i] \in Is[i]\) for every \(i\).
- ChineseRemainderTheorem(I, J, a, b): AlgEtQIdl, AlgEtQIdl, AlgEtQElt, AlgEtQElt -> AlgEtQElt#
Given two coprime ideals \(I\) and \(J\) of \(S\), two elements \(a,b \in S\), finds \(e\) such that \((e-a) \in I\) and \((e-b) \in J\).
- ChineseRemainderTheoremFunctions(Is): SeqEnum[AlgEtQIdl] -> Map, Map#
Given a sequence \(Is\) of \(N\) integral fractional \(S\)-ideals \(I_1,\ldots,I_N\), pairwise coprime, returns a map \(S \to S^N\) representing the natural isomorphism \({S \over I} \to {S \over I_1} \times \cdots \times {S \over I_N}\), where \(I=\prod_i I_i\), and a map \(S^N \to S\) representing the inverse.
- Example: CRT Functions (ex-d7dd94)#
> _<x> := PolynomialRing(Integers()); > A := EtaleAlgebra((x^2+2)*(x^2+3)); > O := MaximalOrder(A); > I1 := PrimesAbove(2*O)[1]; > I2 := PrimesAbove(3*O)[1]; > toProd, fromProd := ChineseRemainderTheoremFunctions([I1,I2]); > toProd(One(A)); [ <1, 0>, <1, 0> ]