Chronological table
Prerequisites and notations
Table of notations
PART Ⅰ.ELEMENTARY THEORY
Chapter Ⅰ.Locally compact fields
1.Finite fields
2.The module in a locally compact field
3.Classification of locally compact fields
4.Structure of p-fields
Chapter Ⅱ.Lattices and duality over local fields
1.Norms
2.Lattices
3.Multiplicative structure of local fields
4.Lattices over R
5.Duality over local fields
Chapter Ⅲ.Places of A-fields
1.A-fields and their completions
2.Tensor-products of commutative fields
3.Traces and norms
4.Tensor-products of A-fields and local fields
Chapter Ⅳ.Adeles
1.Adeles of A-fields
2.The main theorems
3.Ideles
4.Ideles of A-fields
Chapter Ⅴ.Algebraic number-fields
1.Orders in algebras over Q
2.Lattices over algebraic number-fields
3.Ideals
4.Fundamental sets
Chapter Ⅵ.The theorem of Riemann-Roch
Chapter Ⅶ.Zeta-functions of A-fields
1.Convergence of Euler products
2.Fourier transforms and standard functions
3.Quasicharacters
4.Quasicharacters of A-fields
5.The functional equation
6.The Dedekind zeta-function
7.L-functions
8.The coefficients of the L-series
Chapter Ⅷ.Traces and norms
1.Traces and norms in local fields
2.Calculation of the different
3.Ramification theory
4.Traces and norms in A-fields
5.Splitting places in separable extensions
6.An application to inseparable extensions
PART Ⅱ.CLASSFIELD THEORY
Chapter Ⅸ.Simple algebras
1.Structure of simple algebras
2.The representations of a simple algebra
3.Factor-sets and the Brauer group
4.Cyclic factor-sets
5.Special cyclic factor-sets
Chapter Ⅹ.Simple algebras over local fields
1.Orders and lattices
2.Traces and norms
3.Computation of some integrals
Chapter Ⅺ.Simple algebras over A-fields
1.Ramification
2.The zeta-function of a simple algebra
3.Norms in simple algebras
4.Simple algebras over algebraic number-fields
Chapter Ⅻ.Local classfield theory
1.The formalism of class field theory
2.The Brauer group of a local field
3.The canonical morphism
4.Ramification of abelian extensions
5.The transfer
Chapter ⅩⅢ.Global classfield theory
1.The canonical pairing
2.An elementary lemma
3.Hasse's "law of reciprocity"
4.Classfield theory for Q
5.The Hilbert symbol
6.The Brauer group of an A-field
7.The Hilbert p-symbol
8.The kernel of the canonical morpnism
9.The main theorems
10.Local behavior of abelian extensions
11."Classical" classfield theory
12."Coronidis loco"
Notes to the text
Appendix Ⅰ.The transfer theorem
Appendix Ⅱ.W-groups for local fields
Appendix Ⅲ.Shafarevitch's theorem
Appendix Ⅳ.The Herbrand distribution
Index of definitions
