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 54
Pagina 2
... Respect to a Valuation 4.3 . Complete Fields with Discrete Valuation The Multiplicative Structure of a p - adic Number Field Extension of Valuations 4.4 . 4.5 . 4.6 . Finite Extensions of p - adic Number Fields 4.7 . Kummer Extensions ...
... Respect to a Valuation 4.3 . Complete Fields with Discrete Valuation The Multiplicative Structure of a p - adic Number Field Extension of Valuations 4.4 . 4.5 . 4.6 . Finite Extensions of p - adic Number Fields 4.7 . Kummer Extensions ...
Pagina 12
... 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 field K is by ...
... 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 field K is by ...
Pagina 13
... respect to the origin and has volume V ( M ) with V ( M ) ≥ 2 ′′ l . Then M contains at least one point of L distinct from the origin . ( Hasse ( 1979 ) , Chap . 30.2 ) The application of Minkowski's convex body theorem is based on the ...
... respect to the origin and has volume V ( M ) with V ( M ) ≥ 2 ′′ l . Then M contains at least one point of L distinct from the origin . ( Hasse ( 1979 ) , Chap . 30.2 ) The application of Minkowski's convex body theorem is based on the ...
Pagina 14
... respect to the origin . Since the geometrical mean is smaller or equal the arith- metical mean , we have M ≤ N. Now Theorem 1.10 follows from the computa- tion of the volume of M : 2 ′′ 1 - 2π2n " V ( M ) = n ! Example 4. Let K = Q ...
... respect to the origin . Since the geometrical mean is smaller or equal the arith- metical mean , we have M ≤ N. Now Theorem 1.10 follows from the computa- tion of the volume of M : 2 ′′ 1 - 2π2n " V ( M ) = n ! Example 4. Let K = Q ...
Pagina 19
... 1.19 . M ( O ) is a group with respect to the multiplication of modules . The unit element of this group is O and the inverse of me M ( O ) is N ( m ) ̄1gm where gm { gulu e m } . Furthermore N § 1. Orders in Algebraic Number Fields 19.
... 1.19 . M ( O ) is a group with respect to the multiplication of modules . The unit element of this group is O and the inverse of me M ( O ) is N ( m ) ̄1gm where gm { gulu e m } . Furthermore N § 1. Orders in Algebraic Number Fields 19.
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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Parole e frasi comuni
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