Henselian elements

Henselian Implies Large

In this note we show that the quotient field of a domain which is Henselian with respect to a non-trivial ideal is a large field, and give some applications of this fact, using a specialization theorem for ramified covers of the line over (generalized) Krull fields.

Henselian Function Fields

Definable Henselian Valuation Rings

We give model theoretic criteria for the existence of∃∀ and∀∃formulas in the ring language to define uniformly the valuation rings O of models (K,O) of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings O of valued henselian fields (K,O) whose residue class field k is finite, pseudofinite,...

Henselian Elements Franz-viktor Kuhlmann and Josnei Novacoski with an Appendix by Hagen Knaf

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 OF of (F, v) is generated as an OL-algebra by henselian elements. Moreover, we give a list of equivalent conditions under which OF is generated over OL by fini...

Existential ∅-Definability of Henselian Valuation Rings

In [1], Anscombe and Koenigsmann give an existential ∅-definition of the ring of formal power series F [[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields. §