Finite Dimensional Division Algebras Over Fields

Author : Nathan Jacobson
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 55,5 Mb
Release : 2009-12-09
Category : Mathematics
ISBN : 9783642024290

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Finite-Dimensional Division Algebras over Fields by Nathan Jacobson Pdf

Here, the eminent algebraist, Nathan Jacobsen, concentrates on those algebras that have an involution. Although they appear in many contexts, these algebras first arose in the study of the so-called "multiplication algebras of Riemann matrices". Of particular interest are the Jordan algebras determined by such algebras, and thus their structure is discussed in detail. Two important concepts also dealt with are the universal enveloping algebras and the reduced norm. However, the largest part of the book is the fifth chapter, which focuses on involutorial simple algebras of finite dimension over a field.

Author : A.I. Kostrikin,I.R. Shafarevich
Publisher : Springer Science & Business Media
Page : 248 pages
File Size : 52,8 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9783662032350

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Algebra IX by A.I. Kostrikin,I.R. Shafarevich Pdf

The first contribution by Carter covers the theory of finite groups of Lie type, an important field of current mathematical research. In the second part, Platonov and Yanchevskii survey the structure of finite-dimensional division algebras, including an account of reduced K-theory.

Author : Yurj A. Drozd,Vladimir V. Kirichenko
Publisher : Springer Science & Business Media
Page : 260 pages
File Size : 42,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642762444

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Finite Dimensional Algebras by Yurj A. Drozd,Vladimir V. Kirichenko Pdf

This English edition has an additional chapter "Elements of Homological Al gebra". Homological methods appear to be effective in many problems in the theory of algebras; we hope their inclusion makes this book more complete and self-contained as a textbook. We have also taken this occasion to correct several inaccuracies and errors in the original Russian edition. We should like to express our gratitude to V. Dlab who has not only metic ulously translated the text, but has also contributed by writing an Appendix devoted to a new important class of algebras, viz. quasi-hereditary algebras. Finally, we are indebted to the publishers, Springer-Verlag, for enabling this book to reach such a wide audience in the world of mathematical community. Kiev, February 1993 Yu.A. Drozd V.V. Kirichenko Preface The theory of finite dimensional algebras is one of the oldest branches of modern algebra. Its origin is linked to the work of Hamilton who discovered the famous algebra of quaternions, and Cayley who developed matrix theory. Later finite dimensional algebras were studied by a large number of mathematicians including B. Peirce, C.S. Peirce, Clifford, ·Weierstrass, Dedekind, Jordan and Frobenius. At the end of the last century T. Molien and E. Cartan described the semisimple algebras over the complex and real fields and paved the first steps towards the study of non-semi simple algebras.

Author : A.I. Kostrikin,I.R. Shafarevich
Publisher : Springer
Page : 240 pages
File Size : 49,6 Mb
Release : 2014-03-12
Category : Mathematics
ISBN : 3662032368

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Algebra IX by A.I. Kostrikin,I.R. Shafarevich Pdf

The first contribution by Carter covers the theory of finite groups of Lie type, an important field of current mathematical research. In the second part, Platonov and Yanchevskii survey the structure of finite-dimensional division algebras, including an account of reduced K-theory.

Author : Matej Brešar
Publisher : Springer
Page : 227 pages
File Size : 43,6 Mb
Release : 2014-10-14
Category : Mathematics
ISBN : 9783319086934

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Introduction to Noncommutative Algebra by Matej Brešar Pdf

Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.

Author : Nathan Jacobson
Publisher : Courier Corporation
Page : 706 pages
File Size : 53,6 Mb
Release : 2009-07-22
Category : Mathematics
ISBN : 9780486471877

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Basic Algebra II by Nathan Jacobson Pdf

This classic text and standard reference comprises all subjects of a first-year graduate-level course, including in-depth coverage of groups and polynomials and extensive use of categories and functors. 1989 edition.

Author : Paul Moritz Cohn
Publisher : Cambridge University Press
Page : 522 pages
File Size : 53,7 Mb
Release : 1995-07-28
Category : Mathematics
ISBN : 9780521432177

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Skew Fields by Paul Moritz Cohn Pdf

Non-commutative fields (also called skew fields or division rings) have not been studied as thoroughly as their commutative counterparts and most accounts have hitherto been confined to division algebras, that is skew fields finite-dimensional over their centre. Based on the author's LMS lecture note volume Skew Field Constructions, the present work offers a comprehensive account of skew fields. The axiomatic foundation and a precise description of the embedding problem are followed by an account of algebraic and topological construction methods, in particular, the author's general embedding theory is presented with full proofs, leading to the construction of skew fields. The powerful coproduct theorems of G. M. Bergman are proved here as well as the properties of the matrix reduction functor, a useful but little-known construction providing a source of examples and counter-examples. The construction and basic properties of existentially closed skew fields are given, leading to an example of a model class with an infinite forcing companion which is not axiomatizable. The treatment of equations over skew fields has been simplified and extended by the use of matrix methods, and the beginnings of non-commutative algebraic geometry are presented, with a precise account of the problems that need to be overcome for a satisfactory theory. A separate chapter describes valuations and orderings on skew fields, with a construction applicable to free fields. Numerous exercises test the reader's understanding, presenting further aspects and open problems in concise form, and notes and comments at the ends of chapters provide historical background.

Author : John Dauns
Publisher : Unknown
Page : 446 pages
File Size : 48,6 Mb
Release : 1982
Category : Mathematics
ISBN : UOM:39015042079296

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A Concrete Approach to Division Rings by John Dauns Pdf

Author : Abraham Adrian Albert
Publisher : American Mathematical Soc.
Page : 224 pages
File Size : 44,9 Mb
Release : 1939-12-31
Category : Mathematics
ISBN : 9780821810248

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Structure of Algebras by Abraham Adrian Albert Pdf

The first three chapters of this work contain an exposition of the Wedderburn structure theorems. Chapter IV contains the theory of the commutator subalgebra of a simple subalgebra of a normal simple algebra, the study of automorphisms of a simple algebra, splitting fields, and the index reduction factor theory. The fifth chapter contains the foundation of the theory of crossed products and of their special case, cyclic algebras. The theory of exponents is derived there as well as the consequent factorization of normal division algebras into direct factors of prime-power degree. Chapter VI consists of the study of the abelian group of cyclic systems which is applied in Chapter VII to yield the theory of the structure of direct products of cyclic algebras and the consequent properties of norms in cyclic fields. This chapter is closed with the theory of $p$-algebras. In Chapter VIII an exposition is given of the theory of the representations of algebras. The treatment is somewhat novel in that while the recent expositions have used representation theorems to obtain a number of results on algebras, here the theorems on algebras are themselves used in the derivation of results on representations. The presentation has its inspiration in the author's work on the theory of Riemann matrices and is concluded by the introduction to the generalization (by H. Weyl and the author) of that theory. The theory of involutorial simple algebras is derived in Chapter X both for algebras over general fields and over the rational field. The results are also applied in the determination of the structure of the multiplication algebras of all generalized Riemann matrices, a result which is seen in Chapter XI to imply a complete solution of the principal problem on Riemann matrices.

Author : Jean-Pierre Tignol,Adrian R. Wadsworth
Publisher : Springer
Page : 643 pages
File Size : 49,6 Mb
Release : 2015-04-03
Category : Mathematics
ISBN : 9783319163604

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Value Functions on Simple Algebras, and Associated Graded Rings by Jean-Pierre Tignol,Adrian R. Wadsworth Pdf

This monograph is the first book-length treatment of valuation theory on finite-dimensional division algebras, a subject of active and substantial research over the last forty years. Its development was spurred in the last decades of the twentieth century by important advances such as Amitsur's construction of non crossed products and Platonov's solution of the Tannaka-Artin problem. This study is particularly timely because it approaches the subject from the perspective of associated graded structures. This new approach has been developed by the authors in the last few years and has significantly clarified the theory. Various constructions of division algebras are obtained as applications of the theory, such as noncrossed products and indecomposable algebras. In addition, the use of valuation theory in reduced Whitehead group calculations (after Hazrat and Wadsworth) and in essential dimension computations (after Baek and Merkurjev) is showcased. The intended audience consists of graduate students and research mathematicians.

Author : Yoon Sung Hwang
Publisher : Unknown
Page : 270 pages
File Size : 40,8 Mb
Release : 1992
Category : Electronic
ISBN : UCSD:31822005087903

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The Corestriction of Valued Division Algebras Over Henselian Fields by Yoon Sung Hwang Pdf

Author : Anthony W. Knapp
Publisher : Springer Science & Business Media
Page : 730 pages
File Size : 41,6 Mb
Release : 2007-10-11
Category : Mathematics
ISBN : 9780817646134

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Advanced Algebra by Anthony W. Knapp Pdf

Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Advanced Algebra includes chapters on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry. Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems. Together the two books give the reader a global view of algebra and its role in mathematics as a whole.

Author : R.S. Pierce
Publisher : Springer Science & Business Media
Page : 448 pages
File Size : 53,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781475701630

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Associative Algebras by R.S. Pierce Pdf

For many people there is life after 40; for some mathematicians there is algebra after Galois theory. The objective ofthis book is to prove the latter thesis. It is written primarily for students who have assimilated substantial portions of a standard first year graduate algebra textbook, and who have enjoyed the experience. The material that is presented here should not be fatal if it is swallowed by persons who are not members of that group. The objects of our attention in this book are associative algebras, mostly the ones that are finite dimensional over a field. This subject is ideal for a textbook that will lead graduate students into a specialized field of research. The major theorems on associative algebras inc1ude some of the most splendid results of the great heros of algebra: Wedderbum, Artin, Noether, Hasse, Brauer, Albert, Jacobson, and many others. The process of refine ment and c1arification has brought the proof of the gems in this subject to a level that can be appreciated by students with only modest background. The subject is almost unique in the wide range of contacts that it makes with other parts of mathematics. The study of associative algebras con tributes to and draws from such topics as group theory, commutative ring theory, field theory, algebraic number theory, algebraic geometry, homo logical algebra, and category theory. It even has some ties with parts of applied mathematics.

Author : M. Hazewinkel
Publisher : Elsevier
Page : 542 pages
File Size : 50,5 Mb
Release : 2006-05-30
Category : Mathematics
ISBN : 0080462499

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Handbook of Algebra by M. Hazewinkel Pdf

Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest. In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader. An excellent index has been included which is extensive and not limited to definitions, theorems etc. The Handbook of Algebra will publish articles as they are received and thus the reader will find in this third volume articles from twelve different sections. The advantages of this scheme are two-fold: accepted articles will be published quickly and the outline of the Handbook can be allowed to evolve as the various volumes are published. A particularly important function of the Handbook is to provide professional mathematicians working in an area other than their own with sufficient information on the topic in question if and when it is needed. - Thorough and practical source for information - Provides in-depth coverage of new topics in algebra - Includes references to relevant articles, books and lecture notes

Author : Richard D. Schafer
Publisher : Courier Dover Publications
Page : 176 pages
File Size : 44,8 Mb
Release : 2017-11-15
Category : Mathematics
ISBN : 9780486164175

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An Introduction to Nonassociative Algebras by Richard D. Schafer Pdf

Concise graduate-level introductory study presents some of the important ideas and results in the theory of nonassociative algebras. Places particular emphasis on alternative and (commutative) Jordan algebras. 1966 edition.