In 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields.

We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ . This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$ . Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$ -adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

Let $k$ be a locally compact complete field with respect to a discrete valuation $v$ . Let $ \mathcal{O} $ be the valuation ring, $\mathfrak{m}$ the maximal ideal and $F(x)\in \mathcal{O} [x] $ a monic separable polynomial of degree $n$ . Let $\delta = v(\mathrm{Disc} (F))$ . The Montes algorithm computes an OM factorization of $F$ . The single-factor lifting algorithm derives from this data a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$ , for a prescribed precision $\nu $ . In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O({n}^{2+ \epsilon } + {n}^{1+ \epsilon } {\delta }^{2+ \epsilon } + {n}^{2} {\nu }^{1+ \epsilon } )$ word operations for the complexity of the computation of a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$ , assuming that the residue field of $k$ is small.

Multirings are objects like rings but with multi-valued addition. In the present paper we extend results of E. Becker and others concerning orderings of higher level on fields and rings to orderings of higher level on hyperfields and multirings and, in the process of doing this, we establish higher level analogs of the results previously obtained by the second author. In particular, we introduce a class of multirings called ℓ-real reduced multirings, define a natural reflection A ⇝ Qℓ-red(A) from the category of multirings satisfying to the full subcategory of ℓ-real reduced multirings, and provide an elementary first-order description of these objects. The relationship between ℓ-real reduced hyperfields and the spaces of signatures defined by Mulcahy and Powers is also examined.

Let K be a complete ultrametric algebraically closed field and let A be the Banach K-algebra of bounded analytic functions in the ‘open’ unit disc D of K provided with the Gauss norm. Let Mult(A,‖ · ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Multm(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal and let Multa(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. We complete the characterization of continuous multiplicative norms of A by proving that the Gauss norm defined on polynomials has a unique continuation to A as a norm: the Gauss norm again. But we find prime closed ideals that are neither maximal nor null. The Corona Problem on A lies in two questions: is Multa(A, ‖ · ‖) dense in Multm(A, ‖ · ‖)? Is it dense in Multm(A, ‖ · ‖)? In a previous paper, Mainetti and Escassut showed that if each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖), then the answer to the first question is affirmative. In particular, the authors showed that when K is strongly valued each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖). Here we prove that this uniqueness also holds when K is spherically complete, and therefore so does the density of Multa(A, ‖ · ‖) in Multm(A, ‖ · ‖).

Let υ be a Henselian valuation of arbitrary rank of a field K, and let ῡ be the (unique) extension of v to a fixed algebraic closure of K. For an element α ∈ \K, a chain α = α0, α1,…,αr of elements of , such that αi is of minimum degree over K with the property that ῡ(αi−1 − αi) = sup{ῡ(αi−1 − β) | [K (β) : K] < [K (αi−1) : K]} and that αr ∈ K, is called a saturated distinguished chain for α with respect to (K, υ). The notion of a saturated distinguished chain has been used to obtain results about the irreducible polynomials over any complete discrete rank one valued field K and to determine various arithmetic and metric invariants associated to elements of (cf. [J. Number Theory, 52 (1995), 98–118.] and [J. Algebra, 266 (2003), 14–26]). In this paper, a method is described of constructing a saturated distinguished chain for α, and also determining explicitly some invariants associated to α, when the degree of the extension K (α)/K is not divisible by the characteristic of the residue field of υ.

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

Let ν be a rank 1 henselian valuation of a field K having unique extension ῡ to an algebraic closure of K. For any subextension L/K of /K, let G (L), Res (L) denote respectively the value group and the residue field of the valuation obtained by restricting ῡ to L. If a∈\K define

We give a positive answer to a question of Horst Tietz. A theorem of his that is related to the Mittag-Leffler theorem looks like a duality restult under some locally convex topology on the space of meromorphic functions. Tietz has posed the problem of finding such a topology. It is shown that a topology introduced by Holdguün in 1973 solves the problem. The main tool in the study of this topology is a projective description of it that is derived here. We also argue that Holdgrün's topology is the natural locally convex topology on the space of meromorphic functions.

Introduction. Throughout the paper K(x) is a simple transcendental extension of a field K; v is a valuation of K and w is an extension of v to K(x). Also koÍk and GoÍG denote respectively the residue fields and the value groups of the valuations v and w. A well-known theorem conjectured by Nagata asserts that either k is an algebraic extension of feo or k is a simple transcendental extension of a finite extension of ko (cf [4] or [6] or [1, Corollary 2.3]). We prove here an analogous result for the value groups viz. either G/ Go is a torsion group or there exists a subgroup G1 of G containing Go with [G1: Go] > ∞ such that G is the direct sum of G1 and an infinite cyclic group. Incidentally we obtain a description of the valuation w as well as of its residue field in the second case. Thus a characterization of all those extensions w of v to K(x), for which w(K(x)\{0})/Go is not a torsion group, is given. Corresponding to such a valuation w, we define three numbers N, S and T which satisfy the inequality N ≥= ST. This is analogous to the fundamental inequality established by Ohm (cf. [5, 1.2]) for residually transcendental extensions of v to K(x). We also investigate the conditions under which N = ST