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 76
Pagina 13
... implies Theorem 1.9 : Taking in account ≤ a ̄1m , Prop . 1.5 and ( 1.2 ) , ( 1.3 ) one finds [ a ̄1m : ] = N ( a ̄1m ) ̄1 | NK / ¤ ( α ) | 4 2 n ! = N ( m ) π < ( 1 ) * √14 ( 0 ) 1 . ( 1.4 ) such that Hence in each class of CL ( D ) ...
... implies Theorem 1.9 : Taking in account ≤ a ̄1m , Prop . 1.5 and ( 1.2 ) , ( 1.3 ) one finds [ a ̄1m : ] = N ( a ̄1m ) ̄1 | NK / ¤ ( α ) | 4 2 n ! = N ( m ) π < ( 1 ) * √14 ( 0 ) 1 . ( 1.4 ) such that Hence in each class of CL ( D ) ...
Pagina 14
... implies ( 1.5 ) . It remains to choose M. We put M = { 0,1 , ... , 2 ^ „ ) € R " 1⁄2 12,1 + 2 Σ 12 , + 3 Autral thetral < n } . ... , 2 ) v = r1 + 1 This is obviously a convex closed set in R " which is central symmetric with respect to ...
... implies ( 1.5 ) . It remains to choose M. We put M = { 0,1 , ... , 2 ^ „ ) € R " 1⁄2 12,1 + 2 Σ 12 , + 3 Autral thetral < n } . ... , 2 ) v = r1 + 1 This is obviously a convex closed set in R " which is central symmetric with respect to ...
Pagina 17
... implies R ( ) # 0 . 1.4 . The Unit Group of a Real - Quadratic Number Field . In the case of a real - quadratic field K = Q ( √√d ) one has r = 2 and therefore a fundamental system of units consists of one unit . There is a nice ...
... implies R ( ) # 0 . 1.4 . The Unit Group of a Real - Quadratic Number Field . In the case of a real - quadratic field K = Q ( √√d ) one has r = 2 and therefore a fundamental system of units consists of one unit . There is a nice ...
Pagina 19
... implies the existence of x , y e Z with x2 - dy2 = −4 , but ' x2 - dy2` р = 1 , - = - 1 . where a ( ) denotes the Legendre symbol . On the other hand [ * : E。] = 2 for d = 5 ( Example 5 ) . With the notation above let a = b = 1 and ...
... implies the existence of x , y e Z with x2 - dy2 = −4 , but ' x2 - dy2` р = 1 , - = - 1 . where a ( ) denotes the Legendre symbol . On the other hand [ * : E。] = 2 for d = 5 ( Example 5 ) . With the notation above let a = b = 1 and ...
Pagina 23
... implies the following theorem of Euler : A prime p can be represented as the sum of two squares of natural numbers if and only if p = 2 or p = 1 ( mod 4 ) . There is only one such representa- tion.☐ Below we will show that among orders ...
... implies the following theorem of Euler : A prime p can be represented as the sum of two squares of natural numbers if and only if p = 2 or p = 1 ( mod 4 ) . There is only one such representa- tion.☐ Below we will show that among orders ...
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