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.
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ISBN-13
978-3-540-11920-3 (9783540119203)
DOI
10.1007/978-3-642-68816-4
Schweitzer Klassifikation
Notation and Conventions.- I. Survey of Results.- §1. The Background.- §2. The Classgroup.- §3. Ramification and Module Structure.- §4. Resolvents.- §5. L-Functions and Galois Gauss Sums.- §6. Symplectic Root Numbers and the Class UN/K.- §7. Some Problems and Examples.- Notes to Chapter I.- II. Classgroups and Determinants.- §1. Hom-Description.- §2. Localization.- §3. Change in Basefield and Change in Group.- §4. Reduction mod l and Some Computations.- §5. The Logarithm for Group Rings.- §6. Galois Properties of the Determinant.- Notes to Chapter II.- III. Resolvents, Galois Gauss Sums, Root Numbers, Conductors.- §1. Preliminaries.- §2. Localization of Galois Gauss Sums and of Resolvents.- §3. Galois Action.- §4. Signatures.- §5. The Local Main Theorems.- §6. Non-Ramified Base Field Extension.- §7. Abelian Characters, Completion of Proofs.- §8. Module Conductors and Module Resolvents.- Notes to Chapter III.- IV. Congruences and Logarithmic Values.- §1. The Non-Ramified Characteristic.- §2. Proof of Theorem 31.- §3. Reduction Steps for Theorem 30.- §4. Strategy for Theorem 32.- §5. Gauss Sum Logarithm.- §6. The Congruence Theorems.- §7. The Arithmetic Theory of Tame Local Galois Gauss Sums.- Notes to Chapter IV.- V. Root Number Values.- §1. The Arithmetic of Quaternion Characters.- §2. Root Number Formulae.- §3. Density Results.- §4. The Distribution Theorem.- VI. Relative Structure.- §1. The Background.- §2. Galois Module Structure and the Embedding Problem.- §3. An Example.- §4. Generalized Kummer Theory.- §5. The Generalized Class Number Formula and the Generalized Stickelberger Relation.- Literature List.- List of Theorems.- Some Further Notation.
