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 69
Pagina 10
... show in §1.1 that there is one maximal order Ok containing all orders of K , an element a of K belongs to Ok if and only if there are integers a1 , ... , a , for some s such that as + a1as − 1 + k + as 0 . K K ( 1.1 ) Furthermore we ...
... show in §1.1 that there is one maximal order Ok containing all orders of K , an element a of K belongs to Ok if and only if there are integers a1 , ... , a , for some s such that as + a1as − 1 + k + as 0 . K K ( 1.1 ) Furthermore we ...
Pagina 11
... show that O is finitely generated . Let m be a complete module contained in Ok . If mx take a1 ‡ Ox - m , and let ( m , a , ) be the module in K generated by m and a1 . Then D ( ( m , a , ) ) is a proper divisor of D ( m ) . If ( m , α1 ) ...
... show that O is finitely generated . Let m be a complete module contained in Ok . If mx take a1 ‡ Ox - m , and let ( m , a , ) be the module in K generated by m and a1 . Then D ( ( m , a , ) ) is a proper divisor of D ( m ) . If ( m , α1 ) ...
Pagina 12
... shows that a quadratic field is uniquely determined by its in discriminant . This is not true in general for fields ... show in this section that h ( ) is finite . In the application to the theory of quadratic forms ( § 1.6 ) we need ...
... shows that a quadratic field is uniquely determined by its in discriminant . This is not true in general for fields ... show in this section that h ( ) is finite . In the application to the theory of quadratic forms ( § 1.6 ) we need ...
Pagina 16
... show that the rank of ( * ) is > r 1. This is the main problem . Dirichlet solved it by means of his famous pigeon ... shows the idea of the proof in the simplest non trivial situation ( see also Koch ( 1986 ) , 19.7 , for the proof in ...
... show that the rank of ( * ) is > r 1. This is the main problem . Dirichlet solved it by means of his famous pigeon ... shows the idea of the proof in the simplest non trivial situation ( see also Koch ( 1986 ) , 19.7 , for the proof in ...
Pagina 20
... shows mgm = N ( m ) . Furthermore if m1 , m2 ɛ M ( O ) , then ( m1m2 ) g ( m ̧m2 ) = m1gm1m2gm2 = N ( m1 ) N ( m2 ) O . If O , is the order of m , m2 , then 1 ( 1.10 ) ( 1.11 ) ( 1.10 ) and ( 1.11 ) imply ✪ - 1 O1 and N ( m1 ) N ( m2 ) ...
... shows mgm = N ( m ) . Furthermore if m1 , m2 ɛ M ( O ) , then ( m1m2 ) g ( m ̧m2 ) = m1gm1m2gm2 = N ( m1 ) N ( m2 ) O . If O , is the order of m , m2 , then 1 ( 1.10 ) ( 1.11 ) ( 1.10 ) and ( 1.11 ) imply ✪ - 1 O1 and N ( m1 ) N ( m2 ) ...
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