>I have had trouble understanding class numbers. Usually they measure the "failure" of unique factorization. So how does these failures become a special type of

>I have had trouble understanding class numbers. Usually they measure the "failure" of unique factorization. So how does these failures become a special type of number, that multiply to form a group? That is a very good question. A priori the "ideal class group" is just a set. Why do you get a multiplication rule? I don't think there is any particularly simple explanation. You might read, say, Cox's Primes of the form x^2 + ny^2 for more, but nothing comes to mind now which I can explain in a short amount of space. But please know, at least, that you have identified a very important question!