Chinese Remainder Theorem

In this section we will prove the Chinese Remainder Theorem for rings of integers, deduce several surprising and useful consequences, then learn about discriminants, and finally norms of ideals. We will also define the class group of $\O _K$ and state the main theorem about it. The tools we develop here illustrate the power of what we have already proved about rings of integers, and will be used over and over again to prove other deeper results in algebraic number theory. It is essentially to understand everything we discuss in this chapter very well.