Algebraic Number TheorySpringer Science & Business Media, 6 dic 2012 - 269 pagine From the reviews: "... 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 "... Number theory is not easy and quite technical at several places, as the author is able to show in his technically good exposition. The amount of difficult material well exposed gives a survey of quite a lot of good solid classical number theory... Conclusion: for people not already familiar with this field this book is not so easy to read, but for the specialist in number theory this is a useful description of (classical) algebraic number theory." Medelingen van het wiskundig genootschap, 1995 |
Dall'interno del libro
Risultati 1-5 di 39
Pagina 3
... Polynomial 107 2.2 . The First Main Theorem 108 2.3 . The Reciprocity Law 109 2.4 . The Construction of the Ray Class Field 109 2.5 . Algebraic Theory of Complex Multiplication 2.6 . Generalization §3 . Cohomology of Groups 3.1 ...
... Polynomial 107 2.2 . The First Main Theorem 108 2.3 . The Reciprocity Law 109 2.4 . The Construction of the Ray Class Field 109 2.5 . Algebraic Theory of Complex Multiplication 2.6 . Generalization §3 . Cohomology of Groups 3.1 ...
Pagina 8
... polynomials with coefficients in Ok ( §2.3 ) . Dedekind introduces ideals in Ok , defining so one of the most important notions of algebra , and the third method due to Zolotarev and Hensel uses what is nowadays called localization ...
... polynomials with coefficients in Ok ( §2.3 ) . Dedekind introduces ideals in Ok , defining so one of the most important notions of algebra , and the third method due to Zolotarev and Hensel uses what is nowadays called localization ...
Pagina 12
... polynomial of a with respect to Q has integral coefficients . ☐ m 1 m - 1 ... Proposition 1.8 . Let f ( x ) = x + α1x TM -1 + ··· + α € x [ x ] and let a € K with f ( x ) = 0. Then a is an integer of K. □ Example 2. A quadratic number ...
... polynomial of a with respect to Q has integral coefficients . ☐ m 1 m - 1 ... Proposition 1.8 . Let f ( x ) = x + α1x TM -1 + ··· + α € x [ x ] and let a € K with f ( x ) = 0. Then a is an integer of K. □ Example 2. A quadratic number ...
Pagina 26
... polynomials Bi fx ( 5 ) = NL / K = – Στ1 βυχν of BK Στ = 1 β.Χ. k 1 NL / k ( Σ $ = 1 NL / K3 = 1 ẞ , ( x , 5 — Evk ) ) NL / K ( Σs = 1 ẞyxv ) 1 , ... , s , where NL / K means the norm from L ( z , § ) to K ( z , § ) . We have to show ...
... polynomials Bi fx ( 5 ) = NL / K = – Στ1 βυχν of BK Στ = 1 β.Χ. k 1 NL / k ( Σ $ = 1 NL / K3 = 1 ẞ , ( x , 5 — Evk ) ) NL / K ( Σs = 1 ẞyxv ) 1 , ... , s , where NL / K means the norm from L ( z , § ) to K ( z , § ) . We have to show ...
Pagina 27
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Sommario
7 | |
22 | |
3 Dedekind Rings | 28 |
60 | 42 |
69 | 49 |
76 | 56 |
Harmonic Analysis on Local and Global Fields | 63 |
6 Hecke LSeries and the Distribution of Prime Ideals | 70 |
7 Further Results of Class Field Theory | 145 |
Cohomology of Profinite Groups | 151 |
2 Galois Cohomology of Local and Global Fields | 168 |
Abelian Fields | 192 |
3 Iwasawas Theory of IExtensions | 206 |
Artin LFunctions and Galois Module Structure | 219 |
2 Galois Module Structure and Artin Root Numbers | 234 |
Fields Domains and Complexes | 237 |
Class Field Theory | 90 |
2 Complex Multiplication | 107 |
Simple Algebras | 131 |
6 Explicit Reciprocity Laws and Symbols | 137 |
Tables | 245 |
References | 251 |
Author Index 263 | 262 |
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a₁ abelian extension abelian field abelian group algebraic number field arbitrary Artin 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 function G-module G₁ Galois group global fields group G H¹(G H²(G Hasse 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 module morphism natural number norm symbol normal extension normal subgroup number theory polynomial prime divisors prime element prime ideal prime numbers principal ideal pro-p-group profinite group Proposition quadratic r₂ ramified reciprocity law representation resp roots of unity Shafarevich simple algebra subgroup of G trivial U₁ unramified valuation Z/pZ