“Chinese remainder theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Chinese remainder theorem

释义

Chinese remainder theorem

For coprime natural numbers m and n, ℤmn is isomorphic to ℤm×ℤn as a ring. More concretely, there is unique x in the range 0 ≤ x < mn that solves the simultaneous congruences x ≡ a (mod m) and x ≡ b (mod n). The isomorphism from ℤmn to ℤm×ℤn sends x mod mn to (x mod m, x mod n). If u,v are integers such that um + vn = 1 (see bézout's lemma), then the inverse sends (a mod m, b mod n) to vna + umb mod mn.

More generally, if I and J are coprime ideals of a ring R then

This can be proved by applying the first isomorphism theorem to the homomorphism r ↦ (r + I,r + J).

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