We prove that the class of all the rings $\mathbb {Z}/m\mathbb {Z}$ for all $m>1$ is decidable. This gives a positive solution to a problem of Ax asked in his celebrated 1968 paper on the elementary theory of finite fields [1, Problem 5, p. 270]. In our proof, we reduce the problem to the decidability of the ring of adeles $\mathbb {A}_{\mathbb {Q}}$ of $\mathbb {Q}$.

Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for a t-Henselian non-Henselian ordered field elementarily equivalent to a Henselian field with a specified value group.

In this thesis, we study transfer principles in the context of certain Henselian valued fields, namely Henselian valued fields of equicharacteristic $0$ , algebraically closed valued fields, algebraically maximal Kaplansky valued fields, and unramified mixed characteristic Henselian valued fields with perfect residue field. First, we compute the burden of such a valued field in terms of the burden of its value group and its residue field. The burden is a cardinal related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. We show, for instance, that the Hahn field $\mathbb {F}_p^{\text {alg}}((\mathbb {Z}[1/p]))$ is inp-minimal (of burden 1), and that the ring of Witt vectors $W(\mathbb {F}_p^{\text {alg}})$ over $\mathbb {F}_p^{\text {alg}}$ is not strong (of burden $\omega $ ). This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. We show, for instance, that all types over the power series field $\mathbb {R}((t))$ are definable. Similarly, all types over the quotient field of $W(\mathbb {F}_p^{\text {alg}})$ are definable. This extends previous work of Cubides and Delon and of Cubides and Ye.

These distinct results use a common approach, which has been developed recently. It consists of establishing first a reduction to an intermediate structure called the leading term structure, or $\operatorname {\mathrm {RV}}$ -sort, and then of reducing to the value group and residue field. This leads us to develop similar reduction principles in the context of pure short exact sequences of abelian groups.

Abstract prepared by Pierre Touchard.

E-mail: [email protected]

URL: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9

Scanlon [5] proves Ax-Kochen-Ershov type results for differential-henselian monotone valued differential fields with many constants. We show how to get rid of the condition with many constants.

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.

We consider valued fields with a distinguished contractive map as valued modules over the Ore ring of difference operators. We prove quantifier elimination for separably closed valued fields with the Frobenius map, in the pure module language augmented with functions yielding components for a p-basis and a chain of subgroups indexed by the valuation group.

Let $v$ be a henselian valuation of any rank of a field $K$ and let $\bar{v}$ be the unique extension of $v$ to a fixed algebraic closure $\overline{K}$ of $K$. In 2005, we studied properties of those pairs $\left( \theta ,\,\alpha \right)$ of elements of $\overline{K}$ with $\left[ K\left( \theta \right):K \right]\,>\,\left[ K\left( \alpha \right):K \right]$ where $\alpha $ is an element of smallest degree over $K$ such that

$$\bar{v}\left( \theta \,-\,\alpha \right)\,=\,\sup \left\{ \bar{v}\left( \theta \,-\,\beta \right)\,|\,\beta \,\in \,\bar{K},\,\left[ K\left( \beta \right):K \right]\,<\,\left[ K\left( \theta \right):K \right] \right\}\,.$$

Such pairs are referred to as distinguished pairs. We use the concept of liftings of irreducible polynomials to give a different characterization of distinguished pairs.

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.

Let $v$ be a Henselian Krull valuation of a field $K$. In this paper, the authors give some necessary and sufficient conditions for a finite simple extension of $(K,\,v)$ to be defectless. Various characterizations of algebraically maximal valued fields are also given which lead to a new proof of a result proved by Yu. L. Ershov.

We determine conductor exponent, minimal discriminant and fibre type for elliptic curves over discrete valued fields of equal characteristic 3. Along the same lines, partial results are obtained in equal characteristic 2.

We show that the set of values of an additive polynomial in several variables with arguments in a formal Laurent series field over a finite field has the optimal approximation property: every element in the field has a (not necessarily unique) closest approximation in this set of values. The approximation is with respect to the canonical valuation on the field. This property is elementary in the language of valued rings.

Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T. We give a quantifier elimination relative to T for the theory of pairs (ℛ, V) where ℛ ⊨ T and V ≠ ℛ is the convex hull of an elementary substructure of ℛ. We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs with ℛ a model of T and a proper elementary substructure that is Dedekind complete in ℛ. We deduce that the theory of such “tame” pairs is complete.

A corollary of the main result is that if L is a finite-dimensional Galois extension of a field K and if w is a valuation of L extending a valuation v of K, then K is closed in L if and only if all valuations of L extending v are dependent. A further consequence is a generalization of Ostrowski's criterion for a real-valued valuation to be henselian.