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

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? Here is an article explaining why exp(pi*sqrt(163)) = is near an integer: http://mathoverflow.net/questions/4775/why-are-powers-of-exp-pi-sqrt163-almost-integers?rq=1 http://mathoverflow.net/questions/4775/why-are-powers-of-exp... http://math.stackexchange.com/questions/4544/why-is-e-pi-sqrt163-almost-an-integer/4546 http://math.stackexchange.com/questions/4544/why-is-e-pi-sqr... I have met all three of the mathematicians pictured.

impendia 10y ago
>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!