and

Multiplying these two relations yields

Next we prove the general Chinese Remainder Theorem. We will apply this result with in the rest of this chapter.

Each projection is surjective, so to prove that is surjective, it suffices to show that is in the image of , and similarly for the other factors. By Lemma 5.1.3, is coprime to , so there exists and such that . Then maps to in and to 0 in , hence to 0 in for each , since .

William Stein 2012-09-24