Algebraic Number Theory

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Springer Science & Business Media, Jun 29, 2013 - Mathematics - 357 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 1 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 W eber, 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, 1 have intermingled the ideal and idelic approaches without prejudice for either. 1 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
Existence of a conductor for the Artin symbo
200
Class fields
206
CHAPTER XI
213
Local class field theory and the ramification theorem
219
45
226
Induced characters and Lseries contributions
236
CHAPTER XIII
243
A special computation
250

CHAPTER IV
71
31
90
CHAPTER V
99
Lattice points in parallelotopes
110
A volume computation
116
CHAPTER VI
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
Density of primes in arithmetic progressions
166
CHAPTER IX
179
Exponential and logarithm function
185
The global cyclic norm index
193
Application to the BrauerSiegel theorem
260
Other applications
273
Local functional equation
280
Restricted direct products
287
Global computations
297
Tauberian theorem for Dirichlet series
310
57
313
CHAPTER XVI
321
End of the proofs
327
Explicit Formulas
331
The Weil formula
337
The basic sum and the first part of its evaluation
344
62
353
276
355
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