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Preface | ||
Introduction to Global Field Theory | 1 | |
I | Basic Tools and Notations | 7 |
1 | Places of K | 9 |
2 | Embeddings of a Number Field in its Completions | 12 |
3 | Number and Ideal Groups | 21 |
4 | Idele Groups - Generalized Class Groups | 27 |
5 | Reduced Ideles - Topological Aspects | 45 |
6 | Kummer Extensions | 54 |
II | Reciprocity Maps - Existence Theorems | 65 |
1 | The Local Reciprocity Map - Local Class Field Theory | 65 |
2 | Idele Groups in an Extension L/K | 91 |
3 | Global Class Field Theory: Idelic Version | 104 |
4 | Global Class Field Theory: Class Group Version | 125 |
5 | Ray Class Fields - Hilbert Class Fields | 143 |
6 | The Hasse Principle - For Norms - For Powers | 176 |
7 | Symbols Over Number Fields - Hilbert and Regular Kernels | 195 |
III | Abelian Extensions with Restricted Ramification - Abelian Closure | 221 |
1 | Generalities on H[subscript T][superscript S] / H[superscript S] and its Subextensions | 221 |
2 | Computation of A[subscript T][superscript S]:= Gal(H[subscript T][superscript S](p)/K) and T[subscript T][superscript S]:= tor[subscript Z[subscript p]] (A[subscript T][superscript S]) | 240 |
3 | Compositum of the S-split Z[subscript p]-Extensions - The p-Adic Conjecture | 258 |
4 | Structure Theorems for the Abelian Closure of K | 274 |
5 | Explicit Computations in Incomplete p-Ramification | 342 |
6 | Initial Radical of the Z[subscript p] - Extensions | 348 |
7 | The Logarithmic Class Group | 354 |
IV | Invariant Class Groups in p-Ramification - Genus Theory | 361 |
1 | Reduction to the Case of p-Ramification | 362 |
2 | Injectivity of the Transfer Map A[subscript K][superscript ord][approaches]A[subscript L][superscript ord] | 363 |
3 | Determination of (A[subscript L][superscript ord])[superscript G] and (T[subscript L][superscript ord])[superscript G] - p-Rational Fields | 365 |
4 | Genus Theory with Ramification and Decomposition | 375 |
V | Cyclic Extensions with Prescribed Ramification | 407 |
1 | Study of an Example | 408 |
2 | Construction of a Governing Field | 410 |
3 | Conclusion and Perspectives | 434 |
App | Arithmetical Interpretation of H[superscript 2](G[subscript T][superscript S],Z/p[superscript e]Z) | 441 |
1 | A General Approach by Class Field Theory | 442 |
2 | Complete p-Ramification Without Finite Decomposition | 450 |
3 | The General Case - Infinitesimal Knot Groups | 453 |
Bibliography | 467 | |
Index of Notations | 481 | |
General Index | 487 |
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