# Central Simple Algebras and Galois Cohomology PDF Download

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**Author**: Philippe Gille

**Publisher:** Cambridge University Press

**ISBN:** 1107156378

**Category : **Mathematics

**Languages : **en

**Pages : **431

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**Book Description**
The first comprehensive modern introduction to central simple algebra starting from the basics and reaching advanced results.

**Author**: Philippe Gille

**Publisher:** Cambridge University Press

**ISBN:** 1107156378

**Category : **Mathematics

**Languages : **en

**Pages : **431

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**Book Description**
The first comprehensive modern introduction to central simple algebra starting from the basics and reaching advanced results.

**Author**: Philippe Gille

**Publisher:** Cambridge University Press

**ISBN:** 1139458728

**Category : **Mathematics

**Languages : **en

**Pages : **26

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**Book Description**
This book is the first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields. Starting from the basics, it reaches such advanced results as the Merkurjev-Suslin theorem. This theorem is both the culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, but no homological algebra, the book covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi-Brauer varieties, residue maps and, finally, Milnor K-theory and K-cohomology. The last chapter rounds off the theory by presenting the results in positive characteristic, including the theorem of Bloch-Gabber-Kato. The book is suitable as a textbook for graduate students and as a reference for researchers working in algebra, algebraic geometry or K-theory.

**Author**: Gille

**Publisher:**
**ISBN:** 9780521168915

**Category : **
**Languages : **en

**Pages : **
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**Book Description**

**Author**: Skip Garibaldi

**Publisher:** American Mathematical Soc.

**ISBN:** 0821832875

**Category : **Mathematics

**Languages : **en

**Pages : **168

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**Book Description**
This volume is concerned with algebraic invariants, such as the Stiefel-Whitney classes of quadratic forms (with values in Galois cohomology mod 2) and the trace form of etale algebras (with values in the Witt ring). The invariants are analogues for Galois cohomology of the characteristic classes of topology. Historically, one of the first examples of cohomological invariants of the type considered here was the Hasse-Witt invariant of quadratic forms. The first part classifies such invariants in several cases. A principal tool is the notion of versal torsor, which is an analogue of the universal bundle in topology. The second part gives Rost's determination of the invariants of $G$-torsors with values in $H^3(\mathbb{Q}/\mathbb{Z}(2))$, when $G$ is a semisimple, simply connected, linear group. This part gives detailed proofs of the existence and basic properties of the Rost invariant. This is the first time that most of this material appears in print.

**Author**: Grégory Berhuy

**Publisher:** Cambridge University Press

**ISBN:** 1139490885

**Category : **Mathematics

**Languages : **en

**Pages : **328

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**Book Description**
This is the first detailed elementary introduction to Galois cohomology and its applications. The introductory section is self-contained and provides the basic results of the theory. Assuming only a minimal background in algebra, the main purpose of this book is to prepare graduate students and researchers for more advanced study.

**Author**: Skip Garibaldi

**Publisher:** American Mathematical Soc.

**ISBN:** 0821844040

**Category : **Mathematics

**Languages : **en

**Pages : **102

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**Book Description**
This volume concerns invariants of $G$-torsors with values in mod $p$ Galois cohomology--in the sense of Serre's lectures in the book Cohomological invariants in Galois cohomology--for various simple algebraic groups $G$ and primes $p$. The author determines the invariants for the exceptional groups $F_4$ mod 3, simply connected $E_6$ mod 3, $E_7$ mod 3, and $E_8$ mod 5. He also determines the invariants of $\mathrm{Spin}_n$ mod 2 for $n \leq 12$ and constructs some invariants of $\mathrm{Spin}_{14}$. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfister's results on 10- and 12-dimensional quadratic forms and Rost's theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of $\mathrm{Spin}_n$ is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.

**Author**: Skip Garibaldi

**Publisher:** Springer Science & Business Media

**ISBN:** 1441962115

**Category : **Mathematics

**Languages : **en

**Pages : **348

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**Book Description**
Developments in Mathematics is a book series devoted to all areas of mathematics, pure and applied. The series emphasizes research monographs describing the latest advances. Edited volumes that focus on areas that have seen dramatic progress, or are of special interest, are encouraged as well.

**Author**: Martin Kneser

**Publisher:**
**ISBN:**
**Category : **Algebra, Homological

**Languages : **en

**Pages : **348

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**Book Description**

**Author**: David Harari

**Publisher:** Springer Nature

**ISBN:** 3030439011

**Category : **Mathematics

**Languages : **en

**Pages : **336

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**Book Description**
This graduate textbook offers an introduction to modern methods in number theory. It gives a complete account of the main results of class field theory as well as the Poitou-Tate duality theorems, considered crowning achievements of modern number theory. Assuming a first graduate course in algebra and number theory, the book begins with an introduction to group and Galois cohomology. Local fields and local class field theory, including Lubin-Tate formal group laws, are covered next, followed by global class field theory and the description of abelian extensions of global fields. The final part of the book gives an accessible yet complete exposition of the Poitou-Tate duality theorems. Two appendices cover the necessary background in homological algebra and the analytic theory of Dirichlet L-series, including the Čebotarev density theorem. Based on several advanced courses given by the author, this textbook has been written for graduate students. Including complete proofs and numerous exercises, the book will also appeal to more experienced mathematicians, either as a text to learn the subject or as a reference.

**Author**: Grégory Berhuy

**Publisher:** American Mathematical Soc.

**ISBN:** 0821849379

**Category : **Mathematics

**Languages : **en

**Pages : **276

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**Book Description**
Central simple algebras arise naturally in many areas of mathematics. They are closely connected with ring theory, but are also important in representation theory, algebraic geometry and number theory. Recently, surprising applications of the theory of central simple algebras have arisen in the context of coding for wireless communication. The exposition in the book takes advantage of this serendipity, presenting an introduction to the theory of central simple algebras intertwined with its applications to coding theory. Many results or constructions from the standard theory are presented in classical form, but with a focus on explicit techniques and examples, often from coding theory. Topics covered include quaternion algebras, splitting fields, the Skolem-Noether Theorem, the Brauer group, crossed products, cyclic algebras and algebras with a unitary involution. Code constructions give the opportunity for many examples and explicit computations. This book provides an introduction to the theory of central algebras accessible to graduate students, while also presenting topics in coding theory for wireless communication for a mathematical audience. It is also suitable for coding theorists interested in learning how division algebras may be useful for coding in wireless communication.