Let R be any division ring and let 1 be a polynomial, in the indeterminate X, with coefficients in R. Note that the powers of X are always to the right of the coefficients. We denote the set of all such polynomials by R[X]. B. Beck [3] proved the following theorem for the generalized quaternion division algebra; i.e., any division ring of dimension 4 over its center: THEOREM 1. If f(X) is of degree n then f(X) has either infinitely many or at most n zeros in R. Under a reasonable definition of multiplicity Beck also proved: THEOREM 2. Let (c 1, c 2, …, cn ) be a set of pairwise non-conjugate elements of R, and (m 1, …, mN ) positive integers such that Σmi = n = deg f(x).

The well-known Hasse-Brauer-Noether theorem states that a simple algebra with center a number field k splits over k (i.e., is a full matrix algebra) if and only if it splits over the completion of k at every rank one valuation of k. It is natural to ask whether this principle can be extended to a broader class of fields. In particular, we prove here the following extension.

Es sei C eine konvexe Menge eines Frechet-Raumes F mit einer gewissen Trennbarkeitseigenschaft. Ferner sei / eine bzgl. eines Wahrscheinlichkeitsmaßes P integrierbare Frechet-wertige Funktion, die mit Wahrscheinlichkeit l in der konvexen Menge C liegt. Dann liegt P(/), das Integral von/bzgl. P, ebenfalls in C. Ist P(f) ein Extremalpunkt der konvexen Menge C, dann gilt /=P(/) mit Wahrscheinlichkeit 1. Dieses Ergebnis kann benutzt werden, um die Jensensche Ungleichung und die strikte Jensensche Ungleichung für Frechet-wertige Funktion zu beweisen. Spezialisiert auf den Fall F=IR erhalten wir die bekannten Ergebnisse, vgl. etwa [1], ohne einschränkende Voraussetzungen. Gegenbeispiele zeigen, daß für Frechet-wertige Funktionen auf die Trennbarkeitseigenschaft von C nicht verzichtet werden kann und auch gewisse Abschwächungen der Trennbarkeitseigenschaft nicht ausreichen. Die Ergebnisse dürften für die Wahrscheinlichkeitstheorie und Statistik, insbesondere für die Fälle F~IR oder F Banach-Raum, von Interesse sein.

Article Hasse's principle for simple algebras over function fields of curves. I. Algebras of index 2 and 3; curves of genus 0 and 1. was published on January 1, 1978 in the journal Journal für die reine und angewandte Mathematik (volume 1978, issue 299-300).

Prerequisites ad Notation Part One: Arithmetic Theory of Fields I Valuated Fields Valuations Archimedean Valuations Non-Archimedean valuations Prolongation of a complete valuation to a finite extension Prolongation of any valuation to a finite separable extension Discrete valuations II Dedekind Theory of Ideals Dedekind axioms for S Ideal theory Extension fields III Fields of Number Theory Rational global fields Local fields Global fields Part Two: Abstract Theory of Quadratic Forms VI Quadratic Forms and the Orthogonal Group Forms, matrices and spaces Quadratic spaces Special subgroups of On(V) V The Algebras of Quadratic Forms Tensor products Wedderburn's theorem on central simple algebras Extending the field of scalars The clifford algebra The spinor norm Special subgroups of On(V) Quaternion algebras The Hasse algebra VI The Equivalence of Quadratic Forms Complete archimedean fields Finite fields Local fields Global notation Squares and norms in global fields Quadratic forms over global fields VII Hilbert's Reciprocity Law Proof of the reciprocity law Existence of forms with prescribed local behavior The quadratic reciprocity law Part Four: Arithmetic Theory of Quadratic Forms over Rings VIII Quadratic Forms over Dedekind Domains Abstract lattices Lattices in quadratic spaces IX Integral Theory of Quadratic Forms over Local Fields Generalities Classification of lattices over non-dyadic fields Classification of Lattices over dyadic fields Effective determination of the invariants Special subgroups of On(V) X Integral Theory of Quadratic Forms over Global Fields Elementary properties of the orthogonal group over arithmetic fields The genus and the spinor genus Finiteness of class number The class and the spinor genus in the indefinite case The indecomposable splitting of a definite lattice Definite unimodular lattices over the rational integers Bibliography Index Bibliography Index

Let R be a field of rational functions of one variable over a field of constants R0. Dock Sang Rim (6) has proved that the global reciprocity law in exactly the usual sense holds whenever R0 is an absolutely algebraic quasi-fini te field of characteristic not equal to 0: this was known before only when R0 was a finite field. We shall give another proof of Rim's result by means of a noteworthy generalization of the usual global reciprocity law. Namely, let R0 be a finite field and let F be the set of all fields k contained in some fixed Ralg.clos. and of finite degree over R. The reciprocity law states that there exists a family {fk}, k ∈ F, of functions fk: Ck → G(kabel.clos./k) (where Ck is the idèle class group of k) enjoying certain properties such as the norm transfer law.

A field k is called quasi-finite if it is perfect and if Gk≈Ż where Gk is the Galois group of the algebraic closure kc over k and Ż is the completion of the additive group of the rational integers. The classical reciprocity law on the local field with finite residue field is well-known to hold on local fields with quasi-finite residue field ([4] [5]). Thus it is natural to ask if the global reciprocity law should hold in the ordinary sense (see § 1 below) on the function-fields of one variable over quasi-finite field. We consider here two basic prototypes of non-finite quasi-finite fields:

t 9 63) INVARIANT EIGENDISTRIBUTIONS ON SEMISIMPLE LIE GROUPS 123for p

Let ko be any field, OE a normal separable extension of it, and R = R(k 0l fl) the set of all fields of finite degree over fe 0 which are contained in OE.A class formation is a family of groups {E(K) }xei?, indexed by R, together with a family of monomorphisms $K,L: E(K)-+E(L), defined for KC.L, satisfying the Artin-Tate-Kawada axioms [l].These axioms state that every automorphism of a K(E.R which is identity on k Q defines an automorphism of E{K) ; that these automorphisms commute in a reasonable way with the $K,L\ that if k, K(E:R and k is fixed point field for a finite group H of automorphisms of K then ( 1)H denoting the subgroup of E(K) on which H acts simply; and that(2)for all KÇzR, all finite groups G of automorphisms of K/ko, where #G denotes the number of elements of G. From well-known results of J. Tate [2] it follows that (2) is equivalent to

Previous article Next article A Note on Degree-n IndependenceG. WhaplesG. Whapleshttps://doi.org/10.1137/0106021PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout"A Note on Degree-n Independence." Journal of the Society for Industrial and Applied Mathematics, 6(3), pp. 300–301[1] Henry C. Thacher, Jr., Generalization of concepts related to linear dependence, J. Soc. Indust. Appl. Math., 6 (1958), 288–299 10.1137/0106020 MR0095577 0085.01303 LinkISIGoogle Scholar[2] B. L. Van der Waerden, Modern Algebra, Vol. 2, Ungar, New York, 1953, 5– Google Scholar[3] B. L. Van der Waerden, op. cit., vol. 1, pp. 70–73 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Interpolation in Several VariablesHenry C. Thacher, Jr. and W. E. MilneJournal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 1 | 10 July 2006AbstractPDF (810 KB) Volume 6, Issue 3| 1958Journal of the Society for Industrial and Applied Mathematics199-319 History Submitted:05 March 1958Published online:10 July 2006 InformationCopyright © 1958 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0106021Article page range:pp. 300-301ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

Introduction. Witt [5 ] proved that two binary or ternary quadratic forms, over an arbitrary field (of characteristic not 2) are equivalent if and only if they have the same determinant and Hasse invariant. His proof is brief and elegant but uses a lot of the theory of simple algebras. The purpose of this note is to make this fundamental theorem more accessible by giving a short proof using only the general results of modern theory of cohomology of finite groups. Professor Artin (Princeton lectures, 1956) gave such a proof for fields k over which local class field theory holds. For such fields, the group of values of the quadratic norm residue symbol is cyclic of order 2 while for arbitrary fields it may be any abelian group of exponent 2. So our proof is necessarily different from his; still it owes much to his methods.