Nonunique factorization and principalization in number fields

Abstract

Following what is basically Kummer’s relatively neglected approach to nonunique factorization, we determine the structure of the irreducible factorizations of an element n n in the ring of integers of a number field K K . Consequently, we give a combinatorial expression for the number of irreducible factorizations of n n in the ring. When K K is quadratic, we show in certain cases how quadratic forms can be used to explicitly produce all irreducible factorizations of n n .