Algebraic Number TheoryThe 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). |
Contents
| 3 | |
Galois extensions | 15 |
Discrete valuation rings | 25 |
Definitions and completions | 33 |
Polynomials in complete fields | 43 |
Roots of unity | 47 |
Tamely ramified extensions | 55 |
3 | 59 |
Functional Equation of the Zeta Function Heckes Proof 1 The Poisson summation formula | 245 |
A special computation | 250 |
Functional equation | 253 |
Application to the BrauerSiegel theorem | 260 |
Applications to the ideal function | 262 |
Other applications | 273 |
CHAPTER XIV | 275 |
Local additive duality | 276 |
The different and ramification | 63 |
Relations in ideal classes | 71 |
CHAPTER V | 99 |
Lattice points in parallelotopes | 111 |
4 | 119 |
2 | 125 |
The number of ideals in a given class | 131 |
Adeles | 139 |
Generalized ideal class groups relations with idele classes | 145 |
Embedding of k in the idele classes | 151 |
The Lseries | 165 |
Faltings finiteness theorem | 171 |
CHAPTER IX | 179 |
Exponential and logarithm functions | 185 |
The global cyclic norm index | 193 |
Existence of a conductor for the Artin symbol | 200 |
Class fields | 206 |
CHAPTER XI | 213 |
Local class field theory and the ramification theorem | 219 |
Infinite divisibility of the universal norms | 226 |
Induced characters and Lseries contributions | 236 |
Part Three Analytic Theory | 240 |
CHAPTER XIII | 243 |
Local multiplicative theory | 278 |
Local functional equation | 280 |
Local computations | 282 |
Restricted direct products | 287 |
Global additive duality and RiemannRoch theorem | 289 |
Global functional equation | 292 |
Global computations | 297 |
CHAPTER XV | 303 |
Ikeharas Tauberian theorem | 304 |
Tauberian theorem for Dirichlet series | 310 |
Nonvanishing of the Lseries | 312 |
Densities | 315 |
CHAPTER XVI | 321 |
An upper estimate for the residue | 322 |
A lower bound for the residue | 323 |
Comparison of residues in normal extensions | 325 |
End of the proofs | 327 |
Brauers lemma | 328 |
Explicit Formulas | 331 |
The Weil formula | 337 |
The basic sum and the first part of its evaluation | 344 |
| 353 | |
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Common terms and phrases
a₁ abelian extension adele archimedean absolute values Artin map assertion assume automorphism bounded Chapter character class field theory coefficients compact complex numbers concludes the proof conjugate contained converges Corollary cyclic DB/A decomposition group Dedekind ring define degree denote discriminant equal factor group finite extension finite number follows formula Fourier fractional ideal functional equation fundamental domain Galois extension Galois group group G Hence homomorphism ideal class group idele inequality integral closure irreducible polynomial isomorphism kernel L-series lattice Lemma Let f Let K/k maximal ideal module n-th roots non-zero number field number of elements open subgroup p-adic positive integer prime ideal prime number Proposition prove quasi-character quotient field Re(s residue class field roots of unity satisfies splits completely subset surjective tamely ramified Theorem totally ramified trivial unit unramified unramified extension vector whence zeta function