  We give a precise description of how the class group of a number field
measures the failure of unique factorization in its ring of integers.
Specifically, following ideas of Kummer, we determine the structure of all
irreducible factorizations of an element in the ring of integers of a number
field, and give a combinatorial description for the number of such
factorizations. In certain cases, we show how quadratic forms can explicitly
provide all such factorizations.
