chapter  6
28 Pages

Transfer Principles in the Theory of Non-unique Factorizations

ByAlfred Geroldinger, Franz Halter-Koch

It is well-known that in general the ring OK of integers of an algebraic number field is not factorial, and classical philosophy in algebraic number theory states that the class group CK of OK measures the deviation from unique factorization. For a long time the only result justifying this opinion was the classical criterion that OK is factorial if and only if CK = 0. Only in 1960, L. Carlitz [8] proved that |CK | ≤ 2 if and only if OK is half-factorial, that is, any two factorizations of an element have the same length. A systematical investigation of phenomena of nonunique factorizations was initiated by W. Narkiewicz in a series of papers in the sixties and seventies (see [30, Ch. 9] for an overview). One of the most important insights of these investigations was the discovery of the connection between factorization problems in OK and combinatorial problems on zero-sum sequences in CK (see [29] and [31]). In the language we use today, this was the first application of a transfer principle: Factorization properties of OK were investigated by means of

factorization properties of the much simpler monoid B(CK) of all zero-sum sequences of CK .