Algebraic Number Theory

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 354 pages
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The present book gives an exposition of the classical basic algebraic and analytic number theory and supersedes my Algebraic Numbers, including much more material, e. g. the class field theory on which I make further comments at the appropriate place later. For different points of view, the reader is encouraged to read the collec tion of papers from the Brighton Symposium (edited by Cassels-Frohlich), the Artin-Tate notes on class field theory, Weil's book on Basic Number Theory, Borevich-Shafarevich's Number Theory, and also older books like those of Weber, Hasse, Hecke, and Hilbert's Zahlbericht. It seems that over the years, everything that has been done has proved useful, theo retically or as examples, for the further development of the theory. Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more. The point of view taken here is principally global, and we deal with local fields only incidentally. For a more complete treatment of these, cf. Serre's book Corps Locaux. There is much to be said for a direct global approach to number fields. Stylistically, I have intermingled the ideal and idelic approaches without prejudice for either. I also include two proofs of the functional equation for the zeta function, to acquaint the reader with different techniques (in some sense equivalent, but in another sense, suggestive of very different moods).
 

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Contents

CHAPTER
3
Chinese remainder theorem
11
Dedekind rings
18
Explicit factorization of a prime
27
Polynomials in complete fields
41
Unramified extensions
48
CHAPTER III
57
The discriminant
64
The global cyclic norm index
192
Existence of a conductor for the Artin symbo
200
Class fields
206
CHAPTER XI
213
Local class field theory and the ramification theorem
219
Infinite divisibility of the universal norms
225
Artin nonabelian Lseries
232
CHAPTER XIII
243

CHAPTER IV
71
CHAPTER V
99
Lattice points in parallelotopes
110
A volume computation
116
31
117
The Ideal Function
123
The number of ideals in a given class
129
CHAPTER VII
137
Generalized ideal class groups relations with idele classes
145
in the idele classes
151
Zeta function of a number field
159
45
164
Density of primes in arithmetic progressions
166
CHAPTER IX
179
Exponential and logarithm functions
185
A special computation
250
Application to the BrauerSiegel theorem
260
Functional Equation Tates Thesis
275
Local computations
282
Global additive duality and RiemannRoch theorem
289
Global computations
297
Tauberian theorem for Dirichlet series
310
CHAPTER XVI
321
End of the proofs
327
57
329
Explicit Formulas
331
Bibliography
340
Index
351
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