Quadratic forms over formally real fields with eight square classes

Complete classification of formally real fields with 8 square classes with respect to the behaviour of quadratic forms is given. Two fields F and K are equivalent with respect to quadratic forms if the quadratic form schemes of the two fields are isomorphic or in other words, if the Witt rings W(F) and W(K) are isomorphic. It is shown here that for formally real fields with 8 square classes there are exactly seven possible quadratic form schemes and for each of the seven schemes a formally real field with 8 square classes and the given scheme is constructed.     

Field invariants under totally positive quadratic extensions

K = F(√d) is a formally real field and a totally positive quadratic extension of F. A decomposition theorem for quadratic forms in Fed (K) is given. The invariants r(q) and ud(KF) are defined and relations between the invariants β_F(i), β_K(i), ud(F), ud(K), l(F), l(K) are studied, using the theory of quadratic forms.     

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.     

On existence of tame Harrison map

We present here two new criteria for existence of a tame Harrison map of two formally real algebraic function fields over a fixed real closed field of constants. The first criterion (c.f. Theorem 2.5) shows that a square class group isomorphism is a tame Harrison map if it induces an isomorphism of the coproduct rings of residue Witt rings. The other result (c.f. Proposition 3.5) associates a tame Harrison map to an integral quaternion-symbol equivalence.      

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.     

Garlands containing the general linear group over an intermediate field

Let K/k be a finite extension of fields with an intermediate subfield L, and let H = GL_L(K) be the general linear group of all L-linear invertible mappings of the vector space of the field K over L. It is proved that the subgroups lying between GL_K(K)^H and the normalizer of H in G, where G = GL_k(K), form a garland. Bibliography: 4 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 34–41.     

Embedding Ordered Valued Domains into Division Rings

We study some classes of ordered domains that are embeddable in division rings. We prove the ordered version of the Cohn–Lichtman embedding theorem for valued domains. A question of Glass is answered in the negative. Furthermore, we prove that universal enveloping algebras of Lie algebras over formally real fields can be embedded into ordered division rings. The author acknowledges the financial support from the state budget by the Slovenian Research Agency (project no. Z1-9570-0101-06).     

Kernels of covered groups,II

In this paper we continue the study of kernels of covered groups. For the finite elementary abelian case, we find necessary and sufficient conditions in terms of an associated lattice, for the kernel K of a covered group to be a field, or a simple ring, or a semisimple ring. Also, we discuss the case when N is a local ring. Furthermore, starting with the group G = (Z_n)ⁿ, we show how to construct the fields and simple rings which arise from covers of G.     

Triples of long root subgroups

Let G = G(Φ, K) be a Chevalley group over a field K of characteristic ≠ 2. In the present paper, we classify the subgroups of G generated by triples of long root subgroups, two of which are opposite, up to conjugacy. For finite fields, this result is contained in papers by B. Cooperstein on the geometry of root subgroups, whereas for SL (n, K) it is proved in a paper by L. Di Martino and the first-named author. All interesting items of our list appear in deep geometric results on abstract root subgroups and quadratic actions by F. Timmesfeld and A. Steinbach, and also by E. Bashkirov. However, for applications to the groups of type E _l, we need a detailed justification of this list, which we could not extract from the published papers. That is why in the present paper, we produce a direct elementary proof based on the reduction to D ₄, where the question is settled by straightforward matrix calculations. Bibliography: 73 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 54–83.     

Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space

We generalize the Lipschitz constant to fields of affine jets and prove that such a field extends to a field of total domain \mathbbRⁿ{\mathbb{R}^n} with the same constant. This result may be seen as the analog for fields of the minimal Kirszbraun’s extension theorem for Lipschitz functions and, therefore, establishes a link between Kirszbraun’s theorem and Whitney’s theorem. In fact this result holds not only in Euclidean \mathbbRⁿ{\mathbb{R}^n} but also in general (separable or not) Hilbert space. We apply the result to the functional minimal Lipschitz differentiable extension problem in Euclidean spaces and we show that no Brudnyi–Shvartsman-type theorem holds for this last problem. We conclude with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case which was originally studied by Aronsson in the continuous case.     

Splitting Patterns and Invariants of Quadratic Forms

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices i_W(φ_k) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN , we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ?(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ? ? φ ∈ Iⁿ F and give some bounds on m depending on n which assure that they have the same splitting pattern.     

On the normalizers of the unitary subgroup in an integral group ring

In this paper, we investigate the properties of the normalizers of the unitary subgroup u_f(ZG) in an integral group rings. One of our main results is Theorem 2.6 which character¬izes the second normalizer of the unitary subgroup. As a conse¬quence of this theorem, we prove that the second normalizer of u_f(ZG) coincides with the first normalizer when G is a periodic group. Among other results, we give necessary and sufficient conditions for which the unitary subgroup is normal in the unit group when G is periodic and also characterize when all bicyclic units are nontrivial and elements of the normalizer of the unitary subgroup.     

The frattini subgroup of the absolute Galois group of a local field

LetK be a local field,T the maximal tamely ramified extension ofK, F the fixed field inK _sof the Frattini subgroup ofG(K), andJ the compositum of all minimal Galois extensions ofK containingT. The main result of the paper is thatF=J. IfK is a global field andK _solv is the maximal prosolvable extension ofK, then the Frattini group of % MathType!End!2!1!(K _solv/K) is trivial. Partially supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Real-local fields and the canonical operation of the automorphism group on the space of orderings

The object of our investigation is the canonical operation of the automorphism group of a formally real field F on X_F , the space of orderings of F. For a naturally distinguished class of formally real fields, the so-called real-local fields, the Baer-Krull-bijection induces on X_F the structure of a module over the endomorphism ring of the group of archimedean classes of F. We show that Aut F acts on X_F by affinities with respect to that module structure. Subsequently, this “arithmetization” of the operation is exemplarily applied to the question of transitivity (“When can any two orderings of F be transformed into each other by some automorphism of F?"), and to the investigation of the subgroup of Aut F generated by all order automorphism groups of F.     

Every finitely generated regular field extension has a stable transcendence base

A fieldK is called stable if every finitely generaed regular field extensionF/K has a transcendence basex ₁, …,x _n with the following properties: The field extensionF/K(x ₁,…,x _n ) is separable and the Galois hull ofF/K(x ₁,…,x _n ) remains regular overK, i.e.K is algebraically closed in . We prove in this paper thatevery field is stable. This generalizes results from [FJ1] and [GJ] which prove that fields of characteristic 0 and infinite perfect fields are stable, respectively. [G] showed that finite fields are stable in dimension 1, i.e. every finitely generated regular field extension of transcendence degree 1 over a finite field has a stable transcendence base. In the last section of this paper we apply the theorem to the construction of PAC fields with additional properties. A fieldK is called PAC if every absolutely irreducible variety overK has at least oneK-rational point.     

The wedderburn-malcev theorem for comodule algebras

Let H be a Hopf algebra over a field k. Under some assumptions on H we state and prove a generalization of the Wedderburn-Malcev theorem for i7-comodule algebras. We show that our version of this theorem holds for a large enough class of Hopf algebras, such as coordinate rings of completely reducible affine algebraic groups, finite dimensional Hopf algebras over fields of characteristic 0 and group algebras. Some dual results are also included.     

整体函数域素数次循环扩域的类群

设K/F是整体函数域的素数l次循环扩张,F是有理函数域F_q(T)上的有限可分扩域.利用函数域的Conner-Hurrelbrink正合六边形与源于短正合列的正合六边形,本文在l整除与不整除基域F的理想类数的情形下,分别研究函数域K理想类群的Sylow l-子群的结构.同时,利用得到的结果,本文给出了基域F的单位为K中元素norm的若干条件.     

Analytic Continuation of Complex Gauge Fields

The main result of this note treats the problem of unique extension of holomorphic gauge fields across closed subsets of complex Euclidean space, and is based on a corresponding extension theorem for holomorphic vector bundles due to N. P. Buchdahl and the author. Alternatively, let F be a unitary gauge field corresponding to a complex differential form of type (1, 1) (e.g., an anti self-dual Yang–Mills field on a punctured ball in C ²). As a corollary of the main theorem, it is seen that a unique extension of such F , which preserves the curvature type, is obtained if the contraction of F with a holomorphic vector field lies in the image of the ?¯-operator of the associated holomorphic vector bundle.