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PART Ⅰ．ELEMENTARY THEORY 1

Chapter Ⅰ．Locally compact fields 1

1．Finite fields 1

2．The module in a locally compact field 3

3．Classification of locally compact fields 8

4．Structure of p-fields 12

Chapter Ⅱ．Lattices and duality over local fields 24

1．Norms 24

2．Lattices 27

3．Multiplicative structure of local fields 31

4．Lattices over R 35

5．Duality over local fields 38

Chapter Ⅲ．Places of A-fields 43

1．A-fields and their completions 43

2．Tensor-products of commutative fields 48

3．Traces and norms 52

4．Tensor-products of A-fields and local fields 56

Chapter Ⅳ．Adeles 59

1．Adeles of A-fields 59

2．The main theorems 64

3．Ideles 71

4．Ideles of A-fields 75

Chapter Ⅴ．Algebraic number-fields 80

1．Orders in algebras over Q 80

2．Lattices over algebraic number-fields 81

3．Ideals 85

4．Fundamental sets 89

Chapter Ⅵ．The theorem of Riemann-Roch 96

Chapter Ⅶ．Zeta-functions of A-fields 102

1．Convergence of Euler products 102

2．Fourier transforms and standard functions 104

3．Quasicharacters 114

4．Quasicharacters of A-fields 118

5．The functional equation 120

6．The Dedekind zeta-function 127

7．L-functions 130

8．The coefficients of the L-series 134

Chapter Ⅷ．Traces and norms 139

1．Traces and norms in local fields 139

2．Calculation of the different 143

3．Ramification theory 147

4．Traces and norms in A-fields 153

5．Splitting places in separable extensions 158

6．An application to inseparable extensions 159

PART Ⅱ．CLASSFIELD THEORY 162

Chapter Ⅸ．Simple algebras 162

1．Structure of simple algebras 162

2．The representations of a simple algebra 168

3．Factor-sets and the Brauer group 170

4．Cyclic factor-sets 180

5．Special cyclic factor-sets 185

Chapter Ⅹ．Simple algebras over local fields 188

1．Orders and lattices 188

2．Traces and norms 193

3．Computation of some integrals 195

Chapter Ⅺ．Simple algebras over A-fields 202

1．Ramification 202

2．The zeta-function of a simple algebra 203

3．Norms in simple algebras 206

4．Simple algebras over algebraic number-fields 210

Chapter Ⅻ．Local classfield theory 213

1．The formalism of classfield theory 213

2．The Brauer group of a local field 220

3．The canonical morphism 226

4．Ramification of abelian extensions 230

5．The transfer 240

Chapter ⅩⅢ．Global classfield theory 244

1．The canonical pairing 244

2．An elementary lemma 250

3．Hasse's"law of reciprocity" 252

4．Classfield theory for Q 257

5．The Hilbert symbol 260

6．The Brauer group of an A-field 264

7．The Hilbert p-symbol 267

8．The kernel of the canonical morphism 271

9．The main theorems 275

10．Local behavior of abelian extensions 277

11．"Classical"classfield theory 281

12．"Coronidis loco" 288

Notes to the text 292

Appendix Ⅰ．The transfer theorem 295

Appendix Ⅱ．W-groups for local fields 298

Appendix Ⅲ．Shafarevitch's theorem 301

Appendix Ⅳ．The Herbrand distribution 308