| Chronological table |
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xii | |
| Prerequisites and notations |
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xiii | |
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xvii | |
| PART I. ELEMENTARY THEORY |
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1 | (23) |
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1 | (2) |
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The module in a locally compact field |
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3 | (5) |
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Classification of locally compact fields |
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8 | (4) |
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12 | (12) |
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Lattices and duality over local fields |
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24 | (19) |
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24 | (3) |
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27 | (4) |
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Multiplicative structure of local fields |
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31 | (4) |
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35 | (3) |
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Duality over local fields |
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38 | (5) |
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43 | (16) |
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A-fields and their completions |
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43 | (5) |
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Tensor-products of commutative fields |
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48 | (4) |
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52 | (4) |
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Tensor-products of A-fields and local fields |
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56 | (3) |
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59 | (21) |
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59 | (5) |
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64 | (7) |
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71 | (4) |
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75 | (5) |
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80 | (16) |
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Orders in algebras over Q |
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80 | (1) |
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Lattices over algebraic number-fields |
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81 | (4) |
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85 | (4) |
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89 | (7) |
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The theorem of Riemann-Roch |
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96 | (6) |
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Zeta-functions of A-fields |
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102 | (37) |
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Convergence of Euler products |
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102 | (2) |
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Fourier transforms and standard functions |
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104 | (10) |
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114 | (4) |
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Quasicharacters of A-fields |
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118 | (2) |
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120 | (7) |
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The Dedekind zeta-function |
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127 | (3) |
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130 | (4) |
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The coefficients of the L-series |
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134 | (5) |
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139 | (23) |
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Traces and norms in local fields |
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139 | (4) |
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Calculation of the different |
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143 | (4) |
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147 | (6) |
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Traces and norms in A-fields |
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153 | (5) |
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Splitting places in separable extensions |
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158 | (1) |
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An application to inseparable extensions |
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159 | (3) |
| PART II. CLASSFIELD THEORY |
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162 | (26) |
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Structure of simple algebras |
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162 | (6) |
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The representations of a simple algebra |
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168 | (2) |
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Factor-sets and the Brauer group |
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170 | (10) |
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180 | (5) |
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Special cyclic factor-sets |
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185 | (3) |
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Simple algebras over local fields |
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188 | (14) |
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188 | (5) |
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193 | (2) |
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Computation of some integrals |
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195 | (7) |
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Simple algebras over A-fields |
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202 | (11) |
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202 | (1) |
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The zeta-function of a simple algebra |
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203 | (3) |
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206 | (4) |
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Simple algebras over algebraic number-fields |
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210 | (3) |
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213 | (31) |
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The formalism of classfield theory |
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213 | (7) |
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The Brauer group of a local field |
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220 | (6) |
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226 | (4) |
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Ramification of abelian extensions |
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230 | (10) |
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240 | (4) |
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244 | (48) |
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244 | (6) |
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250 | (2) |
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Hasse's ``law of reciprocity'' |
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252 | (5) |
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257 | (3) |
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260 | (4) |
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The Brauer group of an A-field |
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264 | (3) |
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267 | (4) |
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The kernel of the canonical morphism |
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271 | (4) |
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275 | (2) |
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Local behavior of abelian extensions |
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277 | (4) |
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``Classical'' classfield theory |
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281 | (7) |
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288 | (4) |
| Notes to the text |
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292 | (3) |
| Appendix I. The transfer theorem |
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295 | (3) |
| Appendix II. W-groups for local fields |
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298 | (3) |
| Appendix III. Shafarevitch's theorem |
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301 | (7) |
| Appendix IV. The Herbrand distribution |
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308 | |
| Index of definitions |
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