Gauss’ lemma and valuation theory
| Autor: | Pham Ngoc Ánh, M.F. Siddoway |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
Lemma (mathematics) Mathematics::Commutative Algebra Rational root theorem 010102 general mathematics Gauss Unique factorization domain 010103 numerical & computational mathematics Céa's lemma 01 natural sciences Fundamental theorem of arithmetic Mathematics (miscellaneous) Five lemma 0101 mathematics Euclid's lemma Mathematics |
| Zdroj: | Quaestiones Mathematicae. 39:603-609 |
| ISSN: | 1727-933X 1607-3606 |
| DOI: | 10.2989/16073606.2015.1119214 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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