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Cover: 9781475756753 | Local Fields | Jean-Pierre Serre | Taschenbuch | Paperback | viii
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Local Fields
Taschenbuch von Jean-Pierre Serre
Sprache: Englisch

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Beschreibung
The goal of this book is to present local class field theory from the cohomo­ logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field. For example, such fields are obtained by completing an algebraic number field; that is one of the aspects of "localisation". The chapters are grouped in "parts". There are three preliminary parts: the first two on the general theory of local fields, the third on group coho­ mology. Local class field theory, strictly speaking, does not appear until the fourth part. Here is a more precise outline of the contents of these four parts: The first contains basic definitions and results on discrete valuation rings, Dedekind domains (which are their "globalisation") and the completion process. The prerequisite for this part is a knowledge of elementary notions of algebra and topology, which may be found for instance in Bourbaki. The second part is concerned with ramification phenomena (different, discriminant, ramification groups, Artin representation). Just as in the first part, no assumptions are made here about the residue fields. It is in this setting that the "norm" map is studied; I have expressed the results in terms of "additive polynomials" and of "multiplicative polynomials", since using the language of algebraic geometry would have led me too far astray.
The goal of this book is to present local class field theory from the cohomo­ logical point of view, following the method inaugurated by Hochschild and developed by Artin-Tate. This theory is about extensions-primarily abelian-of "local" (i.e., complete for a discrete valuation) fields with finite residue field. For example, such fields are obtained by completing an algebraic number field; that is one of the aspects of "localisation". The chapters are grouped in "parts". There are three preliminary parts: the first two on the general theory of local fields, the third on group coho­ mology. Local class field theory, strictly speaking, does not appear until the fourth part. Here is a more precise outline of the contents of these four parts: The first contains basic definitions and results on discrete valuation rings, Dedekind domains (which are their "globalisation") and the completion process. The prerequisite for this part is a knowledge of elementary notions of algebra and topology, which may be found for instance in Bourbaki. The second part is concerned with ramification phenomena (different, discriminant, ramification groups, Artin representation). Just as in the first part, no assumptions are made here about the residue fields. It is in this setting that the "norm" map is studied; I have expressed the results in terms of "additive polynomials" and of "multiplicative polynomials", since using the language of algebraic geometry would have led me too far astray.
Inhaltsverzeichnis
One Local Fields (Basic Facts).- I Discrete Valuation Rings and Dedekind Domains.- II Completion.- Two Ramification.- III Discriminant and Different.- IV Ramification Groups.- V The Norm.- VI Artin Representation.- Three Group Cohomology.- VII Basic Facts.- VIII Cohomology of Finite Groups.- IX Theorems of Tate and Nakayama.- X Galois Cohomology.- XI Class Formations.- Four Local Class Field Theory.- XII Brauer Group of a Local Field.- XIII Local Class Field Theory.- XIV Local Symbols and Existence Theorem.- XV Ramification.- Supplementary Bibliography for the English Edition.
Details
Erscheinungsjahr: 2013
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Reihe: Graduate Texts in Mathematics
Inhalt: viii
241 S.
62 s/w Illustr.
241 p. 62 illus.
ISBN-13: 9781475756753
ISBN-10: 1475756755
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Serre, Jean-Pierre
Übersetzung: Greenberg, Marvin J.
Auflage: Softcover reprint of the original 1st ed. 1979
Hersteller: Springer New York
Springer US, New York, N.Y.
Graduate Texts in Mathematics
Maße: 235 x 155 x 15 mm
Von/Mit: Jean-Pierre Serre
Erscheinungsdatum: 31.05.2013
Gewicht: 0,394 kg
Artikel-ID: 105580964
Inhaltsverzeichnis
One Local Fields (Basic Facts).- I Discrete Valuation Rings and Dedekind Domains.- II Completion.- Two Ramification.- III Discriminant and Different.- IV Ramification Groups.- V The Norm.- VI Artin Representation.- Three Group Cohomology.- VII Basic Facts.- VIII Cohomology of Finite Groups.- IX Theorems of Tate and Nakayama.- X Galois Cohomology.- XI Class Formations.- Four Local Class Field Theory.- XII Brauer Group of a Local Field.- XIII Local Class Field Theory.- XIV Local Symbols and Existence Theorem.- XV Ramification.- Supplementary Bibliography for the English Edition.
Details
Erscheinungsjahr: 2013
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Reihe: Graduate Texts in Mathematics
Inhalt: viii
241 S.
62 s/w Illustr.
241 p. 62 illus.
ISBN-13: 9781475756753
ISBN-10: 1475756755
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Serre, Jean-Pierre
Übersetzung: Greenberg, Marvin J.
Auflage: Softcover reprint of the original 1st ed. 1979
Hersteller: Springer New York
Springer US, New York, N.Y.
Graduate Texts in Mathematics
Maße: 235 x 155 x 15 mm
Von/Mit: Jean-Pierre Serre
Erscheinungsdatum: 31.05.2013
Gewicht: 0,394 kg
Artikel-ID: 105580964
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