Gauss' lemma and valuation theory. Issue 5 (11th August 2016)

• Record Type: Journal Article
• Authors: Ánh, P.N.; Siddoway, M.F.
• Is Part Of: Quaestiones mathematicae. Volume 39:Issue 5(2016)
• Page Start: 603
• Page End: 609
• Publication Date: 2016-08-11
• Subjects: 13A05 -- 13D05 -- 13F05 -- 06F05; Bezout domains -- Gauss' lemma -- lattice ordered groups; Mathematics -- Periodicals; 510.5
• DOI: 10.2989/16073606.2015.1119214
• Languages: English
• ISSNs: 1607-3606

Abstract

Abstract: Gauss' lemma is not only critically important in showing that polynomial rings over unique factorization domains retain unique factorization; it unifies valuation theory. It figures centrally in Krull's classical construction of valued fields 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 polynomials 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.