Nonunique factorization and principalization in number fields

Martin, Kimball

doi:10.1090/S0002-9939-2011-11053-0

Nonunique factorization and principalization in number fields
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Proc. Amer. Math. Soc. 139 (2011), 3025-3038 Request permission

Abstract:

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

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