Related papers: Subfields of algebraically maximal Kaplansky field…
We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of…
A henselian valued field $K$ is called a tame field if its algebraic closure $\tilde{K}$ is a tame extension, that is, the ramification field of the normal extension $\tilde{K}|K$ is algebraically closed. Every algebraically maximal…
We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed…
A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is…
We conjecture that a $p$-algebra over a complete discrete valued field $K$ contains a totally ramified purely inseparable subfield if and only if it contains a totally ramified cyclic maximal subfield. We prove the conjecture in several…
We prove in arbitrary characteristic that an immediate valued algebraic function field $F$ of transcendence degree 1 over a tame field $K$ is contained in the henselization of $K(x)$ for a suitably chosen $x\in F$. This eliminates…
We show that asymptotic (valued differential) fields have unique maximal immediate extensions. Connecting this to differential-henselianity, we prove that any differential-henselian asymptotic field is differential-algebraically maximal,…
We show in this article that in many cases the subfields of a nondegenerate tame semiramified division algebra of prime power degree over a Henselian valued field are inertial field extensions of the center.
Let $K$ be a complete discrete valued field of characteristic $p$ with residue $k$ which is not necessarily perfect. We prove the Conjecture in \cite{cs} that a $p$-algebra over $K$ contains a totally ramified cyclic maximal subfield if it…
So far there exist just a few results about the uniqueness of maximal immediate valued differential field extensions and about the relationship between differential-algebraic maximality and differential-henselianity; see arXiv:1509.02588,…
We introduce and study the notion of maximal tracial algebras. We prove several results in a general setting based on dual pairs and multiplier pairs. In a special case that X is a Banach space we determine the abelian subalgebras of B(X)…
For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long…
We extend the characterization of extremal valued fields given in \cite{[AKP]} to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that…
We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong…
In this paper, we concern the model theory of finitely ramified henselian valued fields via higher valued hyperfields. Most of all, we provide a number of Ax-Kochen-Ershov Theorems for finitely ramified henselian valued fields relative to…
Building over recent results, we expand the basic theory of algebraic extensions to the realm of superfields -a field with multivalued sum and product-, showing that every superfield has a (unique up to isomorphism) strong algebraic…
We give a brief re-exposition of the theory due to Pauli and Sinclair of ramification polygons of Eisenstein polynomials over p-adic fields, their associated residual polynomials and an algorithm to produce all extensions for a given…
Let $k$ be an arbitrary field. We classify the maximal reductive subgroups of maximal rank in any classical simple algebraic $k$-group in terms of combinatorial data associated to their indices. This result complements [S, 2022], which does…
We develop the theory of ``branch algebras'', which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting…
Refining a constructive combinatorial method due to MacLane and Schilling, we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree. If the value group of the…