Henselian Elements

Abstract

Henselian elements are roots of polynomials which satisfy the conditions of Hensel's Lemma. In this paper we prove that for a finite field extension $(F|L,v)$, if $F$ is contained in the absolute inertia field of $L$, then the valuation ring ${\mathcal{O}}_F$ of $(F,v)$ is generated as an ${\mathcal{O}}_L$-algebra by henselian elements. Moreover, we give a list of equivalent conditions under which ${\mathcal{O}}_F$ is generated over ${\mathcal{O}}_L$ by finitely many henselian elements. We prove that if the chain of prime ideals of ${\mathcal{O}}_L$ is well-ordered, then these conditions are satisfied. We give an example of a finite valued inertial extension $(F|L,v)$ for which ${\mathcal{O}}_F$ is not a finitely generated ${\mathcal{O}}_L$-algebra. We also present a theorem that relates the problem of local uniformization with the theory of henselian elements.

Cited by in corpus (3)

• Characterizing finitely generated fields by a single field axiom
• Essential Finite Generation of Valuation Rings in Characteristic Zero Algebraic Function Fields
• Essential finite generation of extensions of valuation rings