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 92
Pagina 8
... called localization ( § 4 ) . An important part of the theory , in particular the Dirichlet unit theorem , is valid for rings in K with K = Q ( D ) , 1 e O and Ok , called orders of K. O O≤ Orders and modules of elements in K appear in ...
... called localization ( § 4 ) . An important part of the theory , in particular the Dirichlet unit theorem , is valid for rings in K with K = Q ( D ) , 1 e O and Ok , called orders of K. O O≤ Orders and modules of elements in K appear in ...
Pagina 9
... called algebraic if it satisfies an equation of the form with a1 , ... , a , E Q. + an 0 An algebraic number field K is a finite field extension of Q lying in C. It is always of the form K = Q ( a ) with an algebraic number a . The ...
... called algebraic if it satisfies an equation of the form with a1 , ... , a , E Q. + an 0 An algebraic number field K is a finite field extension of Q lying in C. It is always of the form K = Q ( a ) with an algebraic number a . The ...
Pagina 10
... called the integers of K , and any complex number a satisfying an equation of the form ( 1.1 ) is called an integral algebraic number . Orders in a number field K arise in a natural way in connection with modules in K : In this ...
... called the integers of K , and any complex number a satisfying an equation of the form ( 1.1 ) is called an integral algebraic number . Orders in a number field K arise in a natural way in connection with modules in K : In this ...
Pagina 11
... called the discriminant of m and is denoted by d ( m ) . ... , Let ẞ1 , ... , ẞn be a basis of O ( m ) and let A e GL „ ( Q ) such that ᏰᎢ ( 1 , ... , n ) T = A ( Bí , ... , Ba ) . Then the absolute value [ det A of the determinant of ...
... called the discriminant of m and is denoted by d ( m ) . ... , Let ẞ1 , ... , ẞn be a basis of O ( m ) and let A e GL „ ( Q ) such that ᏰᎢ ( 1 , ... , n ) T = A ( Bí , ... , Ba ) . Then the absolute value [ det A of the determinant of ...
Pagina 12
... called the conductor of O ,. K Example 3. For number fields K of degree > 2 the maximal order Ok is not always of the form Z [ a ] . For instance let K = Q ( B ) with 83 + p2 - 2ẞ + 8 = 0 . 2ß Then y ( ẞ + ẞ2 ) / 2 € K and 1 , ẞ , y is ...
... called the conductor of O ,. K Example 3. For number fields K of degree > 2 the maximal order Ok is not always of the form Z [ a ] . For instance let K = Q ( B ) with 83 + p2 - 2ẞ + 8 = 0 . 2ß Then y ( ẞ + ẞ2 ) / 2 € K and 1 , ẞ , y is ...
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