Cyclic Galois Extensions Of Commutative Rings

Author : Cornelius Greither
Publisher : Springer
Page : 155 pages
File Size : 45,9 Mb
Release : 2006-11-15
Category : Mathematics
ISBN : 9783540475392

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Cyclic Galois Extensions of Commutative Rings by Cornelius Greither Pdf

The structure theory of abelian extensions of commutative rings is a subjectwhere commutative algebra and algebraic number theory overlap. This exposition is aimed at readers with some background in either of these two fields. Emphasis is given to the notion of a normal basis, which allows one to view in a well-known conjecture in number theory (Leopoldt's conjecture) from a new angle. Methods to construct certain extensions quite explicitly are also described at length.

Author : Stephen Urban Chase,D. K. Harrison,Alex Rosenberg
Publisher : American Mathematical Soc.
Page : 79 pages
File Size : 54,8 Mb
Release : 1969
Category : Commutative rings
ISBN : 9780821812525

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Galois Theory and Cohomology of Commutative Rings by Stephen Urban Chase,D. K. Harrison,Alex Rosenberg Pdf

Author : John Rognes
Publisher : American Mathematical Soc.
Page : 154 pages
File Size : 48,5 Mb
Release : 2008
Category : Commutative algebra
ISBN : 9780821840764

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Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups by John Rognes Pdf

The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.

Author : Timothy J. Ford
Publisher : American Mathematical Soc.
Page : 637 pages
File Size : 44,5 Mb
Release : 2017-09-26
Category : Associative rings
ISBN : 9781470437701

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Separable Algebras by Timothy J. Ford Pdf

This book presents a comprehensive introduction to the theory of separable algebras over commutative rings. After a thorough introduction to the general theory, the fundamental roles played by separable algebras are explored. For example, Azumaya algebras, the henselization of local rings, and Galois theory are rigorously introduced and treated. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups. The text is accessible to graduate students who have finished a first course in algebra, and it includes necessary foundational material, useful exercises, and many nontrivial examples.

Author : Lindsay Childs
Publisher : American Mathematical Soc.
Page : 225 pages
File Size : 53,5 Mb
Release : 2000
Category : Field extensions (Mathematics).
ISBN : 9780821821312

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Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory by Lindsay Childs Pdf

This book studies Hopf algebras over valuation rings of local fields and their application to the theory of wildly ramified extensions of local fields. The results, not previously published in book form, show that Hopf algebras play a natural role in local Galois module theory. Included in this work are expositions of short exact sequences of Hopf algebras; Hopf Galois structures on separable field extensions; a generalization of Noether's theorem on the Galois module structure of tamely ramified extensions of local fields to wild extensions acted on by Hopf algebras; connections between tameness and being Galois for algebras acted on by a Hopf algebra; constructions by Larson and Greither of Hopf orders over valuation rings; ramification criteria of Byott and Greither for the associated order of the valuation ring of an extension of local fields to be Hopf order; the Galois module structure of wildly ramified cyclic extensions of local fields of degree p and p2; and Kummer theory of formal groups. Beyond a general background in graduate-level algebra, some chapters assume an acquaintance with some algebraic number theory. From there, this exposition serves as an excellent resource and motivation for further work in the field.

Author : Andy R. Magid
Publisher : CRC Press
Page : 266 pages
File Size : 46,8 Mb
Release : 2020-09-10
Category : Mathematics
ISBN : 9781000153361

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Rings, Extensions, and Cohomology by Andy R. Magid Pdf

"Presenting the proceedings of a conference held recently at Northwestern University, Evanston, Illinois, on the occasion of the retirement of noted mathematician Daniel Zelinsky, this novel reference provides up-to-date coverage of topics in commutative and noncommutative ring extensions, especially those involving issues of separability, Galois theory, and cohomology."

Author : Jean-Marie De Koninck,Claude Levesque
Publisher : Walter de Gruyter
Page : 1038 pages
File Size : 55,9 Mb
Release : 1989
Category : Mathematics
ISBN : 3110117916

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Number Theory by Jean-Marie De Koninck,Claude Levesque Pdf

Monumental proceedings (very handsomely produced) of a major international conference. The book contains 74 refereed articles which, apart from a few survey papers of peculiar interest, are mostly research papers (63 in English, 11 in French). The topics covered reflect the full diversity of the current trends and activities in modern number theory: elementary, algebraic and analytic number theory; constructive (computational) number theory; elliptic curves and modular forms; arithmetical geometry; transcendence; quadratic forms; coding theory. (NW) Annotation copyrighted by Book News, Inc., Portland, OR

Author : Stefaan Caenepeel
Publisher : Springer Science & Business Media
Page : 516 pages
File Size : 52,7 Mb
Release : 2002-03-31
Category : Mathematics
ISBN : 1402003463

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Brauer Groups, Hopf Algebras and Galois Theory by Stefaan Caenepeel Pdf

This volume is devoted to the Brauer group of a commutative ring and related invariants. Part I presents a new self-contained exposition of the Brauer group of a commutative ring. Included is a systematic development of the theory of Grothendieck topologies and étale cohomology, and discussion of topics such as Gabber's theorem and the theory of Taylor's big Brauer group of algebras without a unit. Part II presents a systematic development of the Galois theory of Hopf algebras with special emphasis on the group of Galois objects of a cocommutative Hopf algebra. The development of the theory is carried out in such a way that the connection to the theory of the Brauer group in Part I is made clear. Recent developments are considered and examples are included. The Brauer-Long group of a Hopf algebra over a commutative ring is discussed in Part III. This provides a link between the first two parts of the volume and is the first time this topic has been discussed in a monograph. Audience: Researchers whose work involves group theory. The first two parts, in particular, can be recommended for supplementary, graduate course use.

Author : Andy R. Magid
Publisher : CRC Press
Page : 184 pages
File Size : 51,6 Mb
Release : 2014-07-14
Category : Mathematics
ISBN : 9781482208061

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The Separable Galois Theory of Commutative Rings by Andy R. Magid Pdf

The Separable Galois Theory of Commutative Rings, Second Edition provides a complete and self-contained account of the Galois theory of commutative rings from the viewpoint of categorical classification theorems and using solely the techniques of commutative algebra. Along with updating nearly every result and explanation, this edition contains a n

Author : Frank DeMeyer,David Harrison,Rick Miranda
Publisher : Unknown
Page : 63 pages
File Size : 52,8 Mb
Release : 1989
Category : Commutative rings
ISBN : 1470408147

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Quadratic Forms Over Q and Galois Extensions of Commutative Rings by Frank DeMeyer,David Harrison,Rick Miranda Pdf

Author : Lawrence C. Washington
Publisher : Springer Science & Business Media
Page : 504 pages
File Size : 48,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461219347

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Introduction to Cyclotomic Fields by Lawrence C. Washington Pdf

This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat’s Last Theorem, and Iwasawa’s theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott’s proof of the vanishing of Iwasawa’s f-invariant.

Author : Jean-Pierre Tignol,Adrian R. Wadsworth
Publisher : Springer
Page : 643 pages
File Size : 43,7 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 : Gilberto Bini,Flaminio Flamini
Publisher : Springer Science & Business Media
Page : 181 pages
File Size : 44,9 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781461509578

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Finite Commutative Rings and Their Applications by Gilberto Bini,Flaminio Flamini Pdf

Foreword by Dieter Jungnickel Finite Commutative Rings and their Applications answers a need for an introductory reference in finite commutative ring theory as applied to information and communication theory. This book will be of interest to both professional and academic researchers in the fields of communication and coding theory. The book is a concrete and self-contained introduction to finite commutative local rings, focusing in particular on Galois and Quasi-Galois rings. The reader is provided with an active and concrete approach to the study of the purely algebraic structure and properties of finite commutative rings (in particular, Galois rings) as well as to their applications to coding theory. Finite Commutative Rings and their Applications is the first to address both theoretical and practical aspects of finite ring theory. The authors provide a practical approach to finite rings through explanatory examples, thereby avoiding an abstract presentation of the subject. The section on Quasi-Galois rings presents new and unpublished results as well. The authors then introduce some applications of finite rings, in particular Galois rings, to coding theory, using a solid algebraic and geometric theoretical background.

Author : J. Bertoin,F. Martinelli,Y. Peres,Saint-Flour Summer School of Probability Theory
Publisher : Springer Science & Business Media
Page : 308 pages
File Size : 42,6 Mb
Release : 1999-11-17
Category : Mathematics
ISBN : 3540665935

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Ecole D'ete de Probabilites de Saint-Flour by J. Bertoin,F. Martinelli,Y. Peres,Saint-Flour Summer School of Probability Theory Pdf

Lecture Notes in Mathematics This series reports on new developments in mathematical research and teaching - quickly, informally and at a high level. The type of material considered for publication includes 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research Texts which are out of print but still in demand may also be considered. The timeliness of a manuscript is sometimes more important than its form, which might be preliminary or tentative. Details of the editorial policy can be found on the inside front-cover of a current volume. Manuscripts should be submitted in camera-ready form according to Springer-Verlag's specification: technical instructions will be sent on request. TEX macros may be found at: http://www.springer.de/math/authors/b-tex.html Select the version of TEX you use and then click on "Monographs". A subject index should be included. We recommend contacting the publisher or the series editors at an early stage of your project. Addresses are given on the inside back-cover.

Author : Stuart Sui-sheng Wang
Publisher : Unknown
Page : 226 pages
File Size : 51,6 Mb
Release : 1975
Category : Commutative rings
ISBN : CORNELL:31924001153562

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Separable Algebras and Free Cubic Extensions Over Commutative Rings by Stuart Sui-sheng Wang Pdf