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AuthorFrรถhlich, Albrecht. author
TitleGalois Module Structure of Algebraic Integers [electronic resource] / by Albrecht Frรถhlich
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1983
Connect tohttp://dx.doi.org/10.1007/978-3-642-68816-4
Descript X, 266 p. online resource

SUMMARY

In this volume we present a survey of the theory of Galois module structure for rings of algebraic integers. This theory has experienced a rapid growth in the last ten to twelve years, acquiring mathematical depth and significance and leading to new insights also in other branches of algebraic number theory. The decisive take-off point was the discovery of its connection with Artin L-functions. We shall concentrate on the topic which has been at the centre of this development, namely the global module structure for tame Galois extensions of numberfields -in other words of extensions with trivial local module structure. The basic problem can be stated in down to earth terms: the nature of the obstruction to the existence of a free basis over the integral group ring ("normal integral basis"). Here a definitive pattern of a theory has emerged, central problems have been solved, and a stage has clearly been reached when a systematic account has become both possible and desirable. Of course, the solution of one set of problems has led to new questions and it will be our aim also to discuss some of these. We hope to help the reader early on to an understanding of the basic structure of our theory and of its central theme, and to motivate at each successive stage the introduction of new concepts and new tools


CONTENT

Notation and Conventions -- I. Survey of Results -- ยง1. The Background -- ยง2. The Classgroup -- ยง3. Ramification and Module Structure -- ยง4. Resolvents -- ยง5. L-Functions and Galois Gauss Sums -- ยง6. Symplectic Root Numbers and the Class UN/K -- ยง7. Some Problems and Examples -- Notes to Chapter I -- II. Classgroups and Determinants -- ยง1. Hom-Description -- ยง2. Localization -- ยง3. Change in Basefield and Change in Group -- ยง4. Reduction mod l and Some Computations -- ยง5. The Logarithm for Group Rings -- ยง6. Galois Properties of the Determinant -- Notes to Chapter II -- III. Resolvents, Galois Gauss Sums, Root Numbers, Conductors -- ยง1. Preliminaries -- ยง2. Localization of Galois Gauss Sums and of Resolvents -- ยง3. Galois Action -- ยง4. Signatures -- ยง5. The Local Main Theorems -- ยง6. Non-Ramified Base Field Extension -- ยง7. Abelian Characters, Completion of Proofs -- ยง8. Module Conductors and Module Resolvents -- Notes to Chapter III -- IV. Congruences and Logarithmic Values -- ยง1. The Non-Ramified Characteristic -- ยง2. Proof of Theorem 31 -- ยง3. Reduction Steps for Theorem 30 -- ยง4. Strategy for Theorem 32 -- ยง5. Gauss Sum Logarithm -- ยง6. The Congruence Theorems -- ยง7. The Arithmetic Theory of Tame Local Galois Gauss Sums -- Notes to Chapter IV -- V. Root Number Values -- ยง1. The Arithmetic of Quaternion Characters -- ยง2. Root Number Formulae -- ยง3. Density Results -- ยง4. The Distribution Theorem -- VI. Relative Structure -- ยง1. The Background -- ยง2. Galois Module Structure and the Embedding Problem -- ยง3. An Example -- ยง4. Generalized Kummer Theory -- ยง5. The Generalized Class Number Formula and the Generalized Stickelberger Relation -- Literature List -- List of Theorems -- Some Further Notation


Mathematics Algebra Field theory (Physics) Number theory Mathematics Number Theory Field Theory and Polynomials



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