Algebraic Number Theory, Volume 62Springer Science & Business Media, 12 set 1997 - 269 pagine From the reviews of the first printing, published as Volume 62 of the Encyclopaedia of Mathematical Sciences: "... The author succeeded in an excellent way to describe the various points of view under which Class Field Theory can be seen. ... In any case the author succeeded to write a very readable book on these difficult themes." Monatshefte fuer Mathematik, 1994 "... Koch's book is written mostly for non-specialists. It is an up-to-date account of the subject dealing with mostly general questions. Special results appear only as illustrating examples for the general features of the theory. It is supposed that the reader has good general background in the fields of modern (abstract) algebra and elementary number theory. We recommend this volume mainly to graduate studens and research mathematicians." Acta Scientiarum Mathematicarum, 1993 |
Dall'interno del libro
Risultati 1-5 di 44
Pagina 1
... . Normal Extensions 3.8 . Ideals in Algebraic Number Fields 3.9 . Cyclotomic Fields 3.10 . Application to Fermat's Last Theorem I 23 25 27 28 29 30 30 33 36 36 40 41 43 §4 . Valuations 4.1 . Definition and First Properties of.
... . Normal Extensions 3.8 . Ideals in Algebraic Number Fields 3.9 . Cyclotomic Fields 3.10 . Application to Fermat's Last Theorem I 23 25 27 28 29 30 30 33 36 36 40 41 43 §4 . Valuations 4.1 . Definition and First Properties of.
Pagina 2
... Properties of Valuations 4.2 . Completion of a Field with Respect to a Valuation 4.3 . Complete Fields with Discrete Valuation 4.4 . 4.5 . 4.6 . 4.7 . The Multiplicative Structure of a p - adic Number Field Extension of Valuations ...
... Properties of Valuations 4.2 . Completion of a Field with Respect to a Valuation 4.3 . Complete Fields with Discrete Valuation 4.4 . 4.5 . 4.6 . 4.7 . The Multiplicative Structure of a p - adic Number Field Extension of Valuations ...
Pagina 7
... properties of general charac- ter of algebraic number fields and of structures related to them . Special results appear only as examples which illustrate general features of the theory . On the other hand important parts of algebraic ...
... properties of general charac- ter of algebraic number fields and of structures related to them . Special results appear only as examples which illustrate general features of the theory . On the other hand important parts of algebraic ...
Pagina 9
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Pagina 10
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Sommario
Basic Number Theory | 8 |
1 Orders in Algebraic Number Fields | 9 |
2 Rings with Divisor Theory | 22 |
3 Dedekind Rings | 27 |
4 Valuations | 45 |
5 Harmonic Analysis on Local and Global Fields | 63 |
6 Hecke LSeries and the Distribution of Prime Ideals | 70 |
Class Field Theory | 90 |
3 Extensions with Given Galois Groups | 182 |
Abelian Fields | 192 |
1 The Integers of an Abelian Field | 193 |
2 The Arithmetical Class Number Formula | 195 |
3 Iwasawas Theory of ΓExtensions | 206 |
4 padic LFunctions | 211 |
Artin LFunctions and Galois Module Structure | 219 |
1 Artin LFunctions | 222 |
1 The Main Theorems of Class Field Theory | 92 |
2 Complex Multiplication | 107 |
3 Cohomology of Groups | 112 |
4 Proof of the Main Theorems of Class Field Theory | 121 |
5 Simple Algebras | 131 |
6 Explicit Reciprocity Laws and Symbols | 137 |
7 Further Results of Class Field Theory | 145 |
Galois Groups | 150 |
1 Cohomology of Profinite Groups | 151 |
2 Galois Cohomology of Local and Global Fields | 168 |
2 Galois Module Structure and Artin Root Numbers | 234 |
Fields Domains and Complexes | 237 |
Quadratic Residues | 240 |
Locally Compact Groups | 241 |
Bernoulli Numbers | 243 |
Tables | 245 |
251 | |
263 | |
266 | |
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a₁ abelian extension abelian field abelian group algebraic number field arbitrary Artin Artin L-Functions automorphism called Chap class field theory closed subgroup cohomology complex conductor conjecture corresponding cyclic Dedekind ring defined denotes Dirichlet discriminant divisor theory exact sequence Example extension L/K extension of Q finite extension finite group finite normal extension following theorem G-module G₁ Galois group global fields group G H¹(G H²(G Hecke character Hence Hilbert homomorphism ideal class group idele imaginary-quadratic implies induces infinite places integers irreducible isomorphism L-functions Lemma Let G Let L/K Main reference maximal module morphism natural number norm symbol normal extension normal subgroup number theory polynomial prime divisors prime ideal prime number pro-p-group profinite group Proposition quadratic r₂ ramified reciprocity law representation resp roots of unity Shafarevich simple algebra ẞe subgroup of G T-extension trivial U₁ unramified V₁ valuation Z/pZ