Generic Splitting Fields of Central Simple Algebras: Galois Cohomology and Non-Excellence
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Differential Forms, Linked Fields and the $ U $-Invariant
Differential Forms, Linked Fields and the u-Invariant Adam Chapman Department of Computer Science, Tel-Hai Academic College, Upper Galilee, 12208 Israel Andrew Dolphin Department of Mathematics, Ghent University, Ghent, Belgium Abstract We associate an Albert form to any pair of cyclic algebras of prime degree p over a field F with char(F) = p which coincides with the classical Albert form when p = 2. We prove that if every Albert form is isotropic then H4(F) = 0. As a result, we obtain that if F is a linked field with char(F) = 2 then its u-invariant is either 0, 2, 4or 8. Keywords: Differential Forms, Quadratic Forms, Linked Fields, u-Invariant, Fields of Finite Characteristic. 2010 MSC: 11E81 (primary); 11E04, 16K20 (secondary) 1. Introduction Given a field F, a quaternion algebra over F is a central simple F-algebra of degree 2. The maximal subfields of quaternion division algebras over F are quadratic field extensions of F. When char(F) , 2, all quadratic field extensions are separable. When char(F) = 2, there are two types of quadratic field extensions: the separable type which is of the form F[x : x2 + x = α] for some α F λ2 + λ : λ F , and the ∈ \ { 2 ∈ } inseparable type which is of the form F[ √α] for some α F× (F×) . In this case, any quaternion division algebra contains both types of field ext∈ ensions,\ which can be seen by its symbol presentation 2 2 1 [α, β)2,F = F x, y : x + x = α, y = β, yxy− = x + 1 . arXiv:1701.01367v2 [math.RA] 22 May 2017 h i If char(F) = p for some prime p > 0, we let ℘(F) denote the additive subgroup λp λ : λ F . -
Garibaldi S. Cohomological Invariants.. Exceptional Groups And
Cohomological Invariants: Exceptional Groups and Spin Groups This page intentionally left blank EMOIRS M of the American Mathematical Society Number 937 Cohomological Invariants: Exceptional Groups and Spin Groups Skip Garibaldi ÜÌ Ê>Ê««i`ÝÊLÞÊ iÌiÛÊ7°Êvv>® ÕÞÊÓääÊÊUÊÊ6ÕiÊÓääÊÊUÊÊ ÕLiÀÊÎÇÊÃiV`ÊvÊÈÊÕLiÀîÊÊUÊÊ-- ÊääÈxÓÈÈ American Mathematical Society Providence, Rhode Island 2000 Mathematics Subject Classification. Primary 11E72; Secondary 12G05, 20G15, 17B25. Library of Congress Cataloging-in-Publication Data Garibaldi, Skip, 1972– Cohomological invariants : exceptional groups and spin groups / Skip Garibaldi ; with an appendix by Detlev W. Hoffmann. p. cm. — (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 937) “Volume 200, number 937 (second of 6 numbers).” Includes bibliographical references and index. ISBN 978-0-8218-4404-5 (alk. paper) 1. Galois cohomology. 2. Linear algebraic groups. 3. Invariants. I. Title. QA169.G367 2009 515.23—dc22 2009008059 Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. The 2009 subscription begins with volume 197 and consists of six mailings, each containing one or more numbers. Subscription prices for 2009 are US$709 list, US$567 institutional member. A late charge of 10% of the subscription price will be imposed on orders received from nonmembers after January 1 of the subscription year. Subscribers outside the United States and India must pay a postage surcharge of US$65; subscribers in India must pay a postage surcharge of US$95. Expedited delivery to destinations in North America US$57; elsewhere US$160. Each number may be ordered separately; please specify number when ordering an individual number. -
Quadratic Forms and Their Applications
Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999. -
The Descent of Biquaternion Algebras in Characteristic
The descent of biquaternion algebras in characteristic two Demba Barrya,b, Adam Chapmanc, Ahmed Laghribid aFacult´edes Sciences et Techniques de Bamako, BP: E3206 Bamako, Mali bDepartement Wiskunde–Informatica, Universiteit Antwerpen, Belgium cDepartment of Computer Science, Tel-Hai College, Upper Galilee, 12208 Israel dUniversit´ed’Artois, Facult´edes Sciences Jean Perrin, Laboratoire de math´ematiques de Lens EA 2462, rue Jean Souvraz - SP18, 62307 Lens, France Abstract In this paper we associate an invariant to a biquaternion algebra B over a field K with a subfield F such that K/F is a quadratic separable extension and char(F) = 2. We show that this invariant is trivial exactly when B B0 ⊗ K for some biquaternion algebra B0 over F. We also study the behavior of this invariant under certain field extensions and provide several interesting examples. Keywords: Kato-Milne Cohomology, Cohomological invariants, Algebras with involution, Biquaternion algebras 2010 MSC: primary 11E81; secondary 11E04, 16K20, 19D45 1. Introduction Given a central simple algebra C over a field K with a subfield F, we say that C has a descent to F if there exists a central simple algebra C0 over F such that C C0 ⊗ K. When [K : F] = exp(C), one necessary condition for C to have a descent to F is that corK/F(C) ∼Br F, because if C C0⊗K ⊗[K:F] then corK/F(C) = corK/F(C0 ⊗ K) = C0 ∼Br F, where ∼Br denotes the Brauer equivalence. If K/F is a separable quadratic extension and Q is a quaternion algebra over K, then corK/F(Q) ∼Br F is a necessary and sufficient condition for Q to have a descent to F by [1, Chapter X, Theorem 21]. -
Witt Rings and Brauer Groups Under Multiquadratic Extensions. II DANIEL B
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 78, 58-90 (1982) Witt Rings and Brauer Groups under Multiquadratic Extensions. II DANIEL B. SHAPIRO* Department of Mathematics, Ohio State University, Columbus, Ohio 43210 JEAN-PIERRE TIGNOL Institut de MathPtnatiques Pure et AppliquPe, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium AND ADRIAN R. WADSWORTH* Department of Mathematics, University of California, San Diego, LaJolla, California 92093 Communicated by Richard G. Swan Received May 8, 1981 0. INTRODUCTION Let F be a field of characteristic not 2, and suppose M is a multiquadratic extension of F. That is, M/F is a finite abelian extension of exponent 2, so that M = F(@) for some finite subgroup G E F/F’, where $ = F - (0). In studying Brauer groups and products of quaternion algebras, the second author was led to consider the homology groups N,(M/F) of -a certain complex gM,F associated to the extension M/F. This complex appears in [ 11, (3.1); 301 and in (1.1) below. The purpose of the present work is to investigate the first homology group N,(M/F) and to exhibit its close connections with quadratic form theory. The higher homology groups are examined in [ 11,301. The field F is said to have property Pi(n) if N,(M/F) = 1 for every multi- quadratic extension M/F with [M:F] Q 2”. Properties P,(l) and P,(2) always hold, but there are examples [29] of fields F for which P,(3) fails. -
Witt Kernels and Brauer Kernels for Quartic Extensions in Characteristic
WITT KERNELS AND BRAUER KERNELS FOR QUARTIC EXTENSIONS IN CHARACTERISTIC TWO DETLEV W. HOFFMANN AND MARCO SOBIECH Abstract. Let F be a field of characteristic 2 and let E/F be a field extension of degree 4. We determine the kernel Wq(E/F ) of the restriction map WqF → WqE between the Witt groups of nondegenerate quadratic forms over F and over E, completing earlier partial results by Ahmad, Baeza, Mammone and Moresi. We also deduct the corresponding result for the Witt kernel W (E/F ) of the restriction map WF → WE between the Witt rings of nondegenerate symmetric bilinear forms over F and over E from earlier results by the first author. As application, we describe the 2-torsion part of the Brauer kernel for such extensions. 1. Introduction When considering algebraic objects defined over fields such as quadratic forms or central simple algebras, an important problem is to characterize their behavior under field extensions, for example to determine which quadratic forms become hyperbolic or which central simple algebras split over a given extension, in other words to compute the Witt kernel or the Brauer kernel for that extension. The purpose of the present paper is to determine Witt kernels for quartic extensions in characteristic 2 completing earlier results by various authors, and to apply this to the determination of the 2-torsion part of the Brauer kernel. These kernels have been previously computed in characteristic not 2. Also, in all characteristics the kernels have been known for quite some time in the case of quadratic extensions, and they are trivial for odd degree extensions due to Springer’s theorem, so quartic extensions in characteristic 2 are the first case where the determination of these kernels was still incomplete. -
The Group Sk 2 of a Biquaternion Algebra
THE GROUP SK2 OF A BIQUATERNION ALGEBRA BAPTISTE CALMES` Abstract. M. Rost has proved the existence of an exact sequence relating the group SK1 - kernel of the reduced norm for K1 - of a biquaternion algebra D whose center is a field F with Galois cohomology groups of F . In this paper, we relate the group SK2 - kernel of the reduced norm for K2 - of D with Galois cohomology of F through an exact sequence. Introduction To understand the K-theory of central simple algebras, one of the most useful tools is the reduced norm. It is defined for K0, K1 and K2. It has been proved (see [14, Proposition 4]) that it cannot be defined for K3 and satisfy reasonable properties. The definition of the reduced norm is trivial for K0, elementary for K1, but much less elementary for K2. A definition for K2 was given by Suslin in [26, Corollary 5.7], which uses the highly non trivial result that the K-cohomology 0 group H (X, K2) is isomorphic to K2F when X is a complete smooth rational va- riety over the field F (see [26, Corollary 5.6]). The kernel of the reduced norm for Ki, i = 0, 1, 2, is denoted by SKi and is difficult to compute for i = 1, 2 (it is always zero for i = 0). The first result on SK1 was obtained by Wang in 1949. He proved in [34] that SK1A is zero when the index of A is a product of different prime num- bers. Whether SK1 was always zero or not was then known as the Tannaka-Artin problem. -
Scaled Trace Forms of Central Simple Algebras
Scaled trace forms of central simple algebras D. W. Lewis Introduction Any central simple algebra over a field of characteristic not two has a well-defined non-singular quadratic form called the trace form attached to it. This quadratic form was studied in [6] from the viewpoint of the algebraic theory of quadratic forms. In this article we examine a generalization of trace forms to \scaled trace forms" which are defined via the reduced trace map together with scaling by a non-zero element of the algebra. In section 1 of the paper we give our basic definitions and obtain necessary and sufficient conditions for the scaled trace forms to be non-singular. In section 2 we deal with the \split" case, i.e. the case when the algebra is a full matrix algebra over the field. In section 3 we investigate the algebraic invariants of scaled trace forms. We show that the determinant of a scaled trace form is, up to sign, equal to the reduced norm of the scaling element. Also we give formulae for the signature at each ordering when the underlying field is formally real. We cannot calculate the Hasse-Witt invariant in general but we determine it in two special cases only. In sections 4 and 5 we relate scaled trace forms to the transfer homomorphism of quadratic form theory and use them to obtain information about the kernel and image of the extension of scalars homomorphism of Witt rings. We use the terminology and notation of the book of Scharlau [9] and refer the reader to this book for any further background information on quadratic form theory. -
On Hermitian Pfister Forms
On hermitian Pfister forms N. Grenier-Boley, E. Lequeu, M. G. Mahmoudi September 12, 2007 Abstract Let K be a field of characteristic different from 2. It is known that a quadratic Pfister form over K is hyperbolic once it is isotropic. It is also known that the dimension of an anisotropic quadratic form over K belonging to a given power of the fundamental ideal of the Witt ring of K is lower bounded. In this paper, weak analogues of these two statements are proved for hermitian forms over a multiquaternion algebra with invo- lution. Consequences for Pfister involutions are also drawn. An invariant uα of K with respect to a non-zero pure quaternion of a quaternion di- vision algebra over K is defined. Upper bounds for this invariant are provided. In particular an analogue is obtained of a result of Elman and Lam concerning the u-invariant of a field of level at most 2. 1 Introduction Throughout this paper, the characteristic of the base field is supposed to be different from 2. Pfister forms play a fundamental role in the theory of quadratic forms. In the literature, efforts have been made to find analogues of these forms in the framework of central simple algebras with involution or of hermitian forms over such algebras. In the first case, a notion of Pfister involution has been defined by Shapiro (see Subsection 2.4): it is nothing but a central simple algebra endowed with an orthogonal involution which is a tensor product of quaternion algebras with involution. Shapiro has also stated a conjecture which predicts a close relation between Pfister involutions and quadratic Pfister forms (see Conjecture 2.8) which has recently been proved by Becher [2]. -
Outer Automorphisms of Algebraic Groups and a Skolem-Noether Theorem for Albert Algebras
Documenta Math. 917 Outer Automorphisms of Algebraic Groups and a Skolem-Noether Theorem for Albert Algebras Dem Andenken Reinhard B¨orgers gewidmet Skip Garibaldi and Holger P. Petersson Received: June 1, 2015 Revised: March 7, 2016 Communicated by Ulf Rehmann Abstract. The question of existence of outer automorphisms of a simple algebraic group G arises naturally both when working with the Galois cohomology of G and as an example of the algebro-geometric problem of determining which connected components of Aut(G) have rational points. The existence question remains open only for four 3 types of groups, and we settle one of the remaining cases, type D4. The key to the proof is a Skolem-Noether theorem for cubic ´etale subalgebras of Albert algebras which is of independent interest. Nec- essary and sufficient conditions for a simply connected group of outer type A to admit outer automorphisms of order 2 are also given. 2000 Mathematics Subject Classification: Primary 20G41; Secondary 11E72, 17C40, 20G15 Contents 1 Introduction 918 2 Jordan algebras 921 3 Cubic Jordan algebras 924 4 The weak and strong Skolem-Noether properties 930 5 Cubic Jordan algebras of dimension 9 932 Documenta Mathematica 21 (2016) 917–954 918 Skip Garibaldi and Holger P. Petersson 6 Normclassesandstrongequivalence 936 7 Albertalgebras:proofofTheoremB 940 3 8 Outer automorphisms for type D4: proof of Theorem A 945 9 Outer automorphisms for type A 948 1 Introduction An algebraic group H defined over an algebraically closed field F is a disjoint union of connected components. The component H◦ containing the identity element is a normal subgroup in H that acts via multiplication on each of the other components. -
Minimal Forms for Function Fields of Pfister Forms
THÉORIE DES NOMBRES Aimées 1994/95-1995/96 BESANÇON Minimal forms for function fields of Pfister forms D.W. HOFFMANN MINIMAL FORMS FOR FUNCTION FIELDS OF PFISTER FORMS Detlev W. Hoffmann Abstract. Let F be a field of characteristic ^ 2, {/>,} a set of Pfister forms over F and K = •?({/>«}) the iterated function field of the />,'s over F. We investigate tbe isotropy behaviour of quadratic forms over F under this field extension. Elman- Lam-Wadsworth have shown that if the Hasse number û of F is < 2n, n = 1,2, and if ail the />,• are of fold > n, then K/F is excellent and the Witt kernel W(K/F) is generated by {/>,}. We extend and generalize these résulta. We show, for example, that if û(F) < 6 or if F is linked and ail the p.-'s are of fold > 3, then K/F is excellent. More generally, if û(F) < 2" and ail the /J,'S are of fold > n + 1 (resp. > n), then K/F is excellent (resp. W(K/F) is generated by {p,}). We also investigate so-called if-minimal forms, i.e. anisotropic forms over F which become isotropic over K but no proper subform does. In two examples, one over R(<) and one over a formally real global field, we use the methods developed in this paper to compute the if-minimal forms explicitly. 1 Introduction Let F be a field of characteristic ^ 2. In this paper we want to investigate the behaviour of quadratic forms over F over extensions K/F where if is a function field of the form F({/>;}) of Pfister forms {/>,}. -
BIQUATERNION DIVISION ALGEBRAS OVER RATIONAL FUNCTION FIELDS 1. Introduction Let E Be a Field of Characteristic Different from 2
BIQUATERNION DIVISION ALGEBRAS OVER RATIONAL FUNCTION FIELDS KARIM JOHANNES BECHER Abstract. Let E be a field which is the center of a quaternion division algebra and which is not real euclidean. Then there exists a biquaternion division alge- bra over the rational function field E(t) which does not contain any quaternion algebra defined over E. The proof is based on the study of Bezoutian forms developed in [1]. Keywords: Milnor K-theory, quadratic form, valuation, ramification, Be- zoutian form Classification (MSC 2010): 12E30, 12G05, 12Y05, 19D45 1. Introduction Let E be a field of characteristic different from 2. Let E(t) denote the rational function field over E, where t is an indeterminate. It is easy to show that there exists a quaternion division algebra over E(t) if and only if E has some field extension of even degree. The purpose of this article is to provide sufficient conditions to have that there exists a biquaternion division algebra over E(t). In [6] and [4, Sections 3 and 4] such examples were given in the case where E is a local number field. In this article a different approach will be presented that uses the study of ramification sequences via associated Bezoutian forms developped in [1]. For a, b ∈ E× consider the biquaternion algebra 2 2 B = t +(a + 1)t + a, a ⊗E(t) t + at + a, ab over E(t). We will obtain by Theorem 4.2 that, for having that B is a division algebra, it suffices that the E-quaternion algebra (a, b) is non-split and that ab, (a − 4)b∈ / E×2.