Albrecht Fröhlich

Galois module structure of algebraic integers

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.

Gauss sums and p-adic division algebras

Arithmetic of local division algebras.- to Gauss Sums.- Functional equation.- One-dimensional representations.- The basic correspondence.- The basic inductive step.- The general inductive process.- Representations of certain group extensions.- Trace computations.- Induction constants for Galois Gauss sums.- Synthesis of results.- modified correspondences.

Classgroups and Hermitian Modules

I Preliminaries.- 1 Locally free modules and locally freely presented torsion modules.- 2 Determinants and the Hom language for classgroups.- 3 Supplement at infinity.- II Involution algebras and the Hermitian classgroup.- 1 Involution algebras and duality.- 2 Hermitian modules.- 3 Pfaffians of matrices.- 4 Pfaffians of algebras.- 5 Discriminants and the Hermitian classgroup.- 6 Some homomorphisms.- 7 Pulling back discriminants.- 8 Unimodular modules.- 8 Products.- III Indecomposable involution algebras.- 1 Dictionary.- 2 The map P.- 3 Discriminants once more.- 4 Norms of automorphisms.- 5 Unimodular classes once more.- IV Change of order.- 1 Going up.- 2 Going down.- V Groups.- 1 Characters.- 2 Character action. Ordinary theory.- 3 Character action. Hermitian theory.- 4 Special formulae.- 5 Special properties of the group ring involution.- 6 Some Frobenius modules.- 7 Some subgroups of the adelic Hermitian classgroup.- VI Applications in arithmetic.- 1 Local theory.- 2 The global discriminant.- Literature.- List of Theorems.- Some further notation.

The Rational Characterization of Certain Sets of Relatively Abelian Extensions

Let H be a class group— in the sense of class-field theory— in the rational field P, whose order is some power of a prime l. With H there is associated an Abelian extension K of P. The purpose of this paper is to determine in rational terms and for all fields K given in the described manner, the set T(K/P) of cyclic extensions A of K of relative degree l, which are absolutely normal. In particular we shall find the ramification laws for these fields A, and the possible extension types of a group of order l by the Galois group of K, which are realized in Galois groups of fields in T(K/P). It is fundamental to the programme outlined, that we aim at obtaining purely rational criteria of determination.