Gauss' Lemma and valuation theory

Abstract

Gauss' lemma is not only critically important in showing that polyno- mial rings over unique factorization domains retain unique factorization; it unies valuation theory. It gures centrally in Krull's classical construction of valued elds with pre-described value groups, and plays a crucial role in our new short proof of the Ohm-Jaffard-Kaplansky theorem on Bezout domains with given lattice-ordered abelian groups. Furthermore, Eisenstein's criterion on the irreducibility of polyno- mials as well as Chao's beautiful extension of Eisenstein's criterion over arbitrary domains, in particular over Dedekind domains, are also obvious consequences of Gauss' lemma. We conclude with a new result which provides a Gauss' lemma for Hermite rings.

Mathematics Subject Classication (2010): Primary 13A05, 13D05, 13F05; Secondary
06F05.

Key words: Bezout domains, Gauss' lemma, lattice ordered groups.