An isomorphism theorem for henselian algebraic extensions of valued fields

In general, the value groups and the residue fields do not suffice to classify the algebraic henselian extensions of a valued fieldK, up to isomorphism overK. We define a stronger, yet natural structure which carries information about additive and multiplicative congruences in the valued field, extending the information carried by value groups and residue fields. We discuss the cases where these “mixed structures” give a solution of the classification problem.     

On finite intersections of “henselian valued” fields

In this note we study finite intersections of henselian valued fields, i.e. fields carrying either a henselian valuation ring or a henselian absolute value which is (real-) archimedean. To be more precise, we intersect a finite number of henselian valued respectively real closed fields such that the induced valuation rings respectively orderings generate different V-topologies on the intersection, and investigate its algebraic and valuation-theoretic properties.     

Symmetric Elements,Hermitian Forms,and Cyclic Involutions

Let k be a field, K/k be a quadratic separable field extension, and 𝒜 a finite dimensional central simple algebra over K. If k is global or the field of fractions of a two-dimensional excellent henselian local domain with an algebraically closed residue field of characteristic zero and the degree of 𝒜 is odd, we prove that all K/k-involutions on 𝒜 are cyclic.     

Rami globali e parametrizzazione

Summary In this paper, using the theory of henselian rings and in particular the notion of henselian and strict henselian couples, we study the behaviour of analytic branches of an affine algebraic variety at a point x, when x varies along a singular subvariety; moreover we construct an algebraic and canonical procedure, which parametrises the branches.

---

---

Lavoro eseguito nell'ambito della sez. 3 del G.N.S.A.G.A. del C.N.R. e con il supporto finanziario del Ministero della Pubblica Istruzione.     

Torseurs sous une variété abélienne sur R((t))

Let V be an henselian discrete valuation ring with real closed residue field and let k be its quotient ring; we denote by k ⁺ and k ⁻ the two real closures of k. Consider a k-abelian variety A. We compute the Galois-cohomology group H ¹(k,A) in terms of the reduction of the dual variety of A and of the semi-algebraic topology of A(k ⁺) and A(k ⁻). The tools we need are Ogg's results concerning valuation rings with algebraically closed residue field, Hochschild–Serre spectral sequence and Scheiderer's local-global principles. At the end we study more precisely the case of an elliptic curve. Received: 23 October 2000     

Fields with two incomparable henselian valuation rings

Let K be a field and C, C' be two incomparable valuation rings of the separable closure of K, Theorem 1.2 states that the intersection of the decomposition groups of C, C', with respect to K, is precisely the inertia group of the composition ring C·C'. We apply this theorem in the study of two special cases of valued fields (L,B). In the first case, B is henselian and there is a subfield K of L such that L|K is a normal extension and B K is not henselian. The second case is that in which B has exactly two prolongations in the separable closure of L. We call these rings semihenselian rings, and they are characterized through Theorems 2.6 and 2.12.This paper is part of author's doctoral dissertation. Financial support for this research was provided by CNP_q (National Research Council) and by Universidade Estadual de Campinas.     

The relative approximation degree in valued function fields

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 transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author’s proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.     

Hensel赋值域

1°Hensel赋值域的理论起源于Kurt Hensel在他的p-进数论中所建立的一条引理,这条引理是对p-进数域上多项式的可分解性所作的一个判别法则。在赋值论出现以后,Rychlik把它推广到一阶赋值的完全域上,从而在赋值理论中引进了一个重要的代数方法。三十年代,Krull对任意阶的赋值引入了完全性的概念,这个重要的引理又在任意阶的完全域上建立起来。但是,这个引理在赋值理论中的重要性并不依赖于赋值域的完全性,首先认识到这个事实的是瑞士数学家Alexander Ostronwski.现在我们遵循他的方式来作如下的规定:     

The principal axis theorem for holomorphic functions

An algebraic approach to Rellich's theorem is given which states that any analytic family of matrices which is normal on the real axis can be diagonalized by an analytic family of matrices which is unitary on the real axis. We show that this result is a special version of a purely algebraic theorem on the diagonalization of matrices over fields with henselian valuations.

     

Reformulation of Hensel’s Lemma and extension of a theorem of Ore

In this paper, we extend the theorem of Ore regarding factorization of polynomials over p-adic numbers to henselian valued fields of arbitrary rank thereby generalizing the main results of Khanduja and Kumar (J Pure Appl Algebra 216:2648–2656, 2012) and Cohen et al. (Mathematika 47:173–196, 2000). As an application, we derive the analogue of Dedekind’s Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in Khanduja and Kumar (Int J Number Theory 4:1019–1025, 2008). The generalized version of Ore’s Theorem leads to an extension of a result of Weintraub dealing with a generalization of Eisenstein Irreducibility Criterion (cf. Weintraub in Proc Am Math Soc 141:1159–1160, 2013). We also give a reformulation of Hensel’s Lemma for polynomials with coefficients in henselian valued fields which is used in the proof of the extended Ore’s Theorem and was proved in Khanduja and Kumar (J Algebra Appl 12:1250125, 2013) in the particular case of complete rank one valued fields.     

Henselianity in the language of rings

We consider four properties of a field K related to the existence of (definable) henselian valuations on K and on elementarily equivalent fields and study the implications between them. Surprisingly, the full pictures look very different in equicharacteristic and mixed characteristic.     

On factorization of polynomials in henselian valued fields

Guàrdia, Montes and Nart generalized the well-known method of Ore to find complete factorization of polynomials with coe?cients in finite extensions of p-adic numbers using Newton polygons of higher order (cf. [Trans. Amer. Math. Soc. 364 (2012), 361–416]). In this paper, we develop the theory of higher order Newton polygons for polynomials with coe?cients in henselian valued fields of arbitrary rank and use it to obtain factorization of such polynomials. Our approach is different from the one followed by Guàrdia et al. Some preliminary results needed for proving the main results are also obtained which are of independent interest.     

Anelli Henseliani topologici

Summary In the present work we give a generalization of the concept of ? ring henselian with respect to its idealm ?, by introducing the concept of ? ring henselian with respect to the idealm and the linear topology τ ?. Then we get the henselization of a triple (A,m, τ) (ring, ideal, linear topology) and investigate its relations with completion, mainly in the ?m-adic ? situation. Among our results there is also a reformulation, with less restrictive hypothesis, of the Hensel lemma as it is given in Bourbaki.

---

---

Entrata in Redazione il 24 marzo 1971.

---

Lavoro eseguito nell'ambito dei Contratti di ricerca del Comitato Nazionale per la matematica del C.N.R.     

Stable reduction of curves and tame ramification

We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain a new proof of Saito’s criterion, avoiding the use of ℓ-adic cohomology and vanishing cycles.     

Anelli Henseliani topologici - II

Summary In the present work we study some problems about henselian triples and henselization which have been introduced in our article [18]. Mainly we prove that henselian triples coincide with strong henselian triples and give a new formulation of the Hensel lemma, stronger than that we gave in [18]. Then we investigate some properties of henselian triples (changement of ideal or of topology, ecc.) and prove commutativity with quotient.

---

---

Lavoro eseguito nell'ambito dei contratti di ricerca del Comitato Nazionale per la Matematica del C.N.R.

---

Entrata in Redazione il 27 maggio 1971.     

A Note on Class Field Theory for Two-Dimensional Local Rings

The class field theory for the fraction field of a two-dimensional complete normal local ring with finite residue field is established by S. Saito. In this paper, we investigate the index of the norm group in the K ₂-idele class group for a finite Abelian extension of such fields and deduce that the existence theorem does not hold for almost fields in this case.     

Approximation for Frobenius algebraic equations in Witt vectors

We prove an approximation property for solutions to difference equations in excellent discrete valuation rings satisfying an appropriate Hensel's lemma, analog to a theorem of Greenberg [M. Greenberg, Rational points in henselian discrete valuation rings, Publ. Math. Inst. Hautes Études Sci. 31 (1966) 59–64]. In the case of Witt vectors we obtain a Nullstellensatz for Frobenius algebraic equations.     

Closed stability index of excellent henselian local rings

We show that the closed stability index of an excellent henselian local ring of real dimension d2 with real closed residue field is When d=2 it is shown that the value of can be either 2 or 3 and give characterizations of each of these values in terms of the relation of A with its normalization and in terms of the real spectrum of A.Mathematics Subject Classification (2000): 14P15, 32B10, 13J15, 13J25Partially supported by DGES BFM2002-04797 and EC contract HPRN-CT-2001-00271in final form: 15 October 2003     

Maps on Ultrametric Spaces,Hensel's Lemma,and Differential Equations Over Valued Fields

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multidimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.