The Theory of Numbers

Preface
Contents
Table of Notations
CHAPTER I. Cohomology of Groups
	§1. Tensor products and groups of homomorphisms
		1.1. g-modules
		1.2. Tensor products
		1.3. Groups of homomorphisms
	§2. Homology and cohomology
		2.1. Homology of complexes
		2.2. Coefficient modules
		2.3. Homology and cohomology of groups
		2.4. The Duality Theorem
	§3. Inhomogeneous complexes \mathscr{C}( \mathfrak{g} )
		3.1. Definitions
		3.2. The first homology and cohomology groups
		3.3. Extensions of groups and the second cohomology groups
	§4. Subgroups and related cohomology groups
		4.1. Homomorphisms of groups and their cohomology theory
		4.2. Cohomology groups of conjugate subgroups
		4.3. Restriction and transfer
	§5. Tensor products and cup products
		5.1. Direct products of groups and their cohomology theory
		5.2. Cup products
		5.3. Cup products and other mappings
	§6. Cohomology theory of finite groups
		6.1. Dual modules
		6.2. Cohomology groups of negative degrees
		6.3. Restriction and transfer
		6.4. Cup products
	§7. Cohomology theory of cyclic groups
		7.1. Monomial complexes
		7.2. The Herbrand quotient
	§8. Tate's Theorem and Galois cohomology
		8.1. Tate's Theorem
		8.2. Galois cohomology
CHAPTER II. Valuation Theory
	§1. Valuations of fields
		1.1. Valuations
		1.2. Non-Arcbimedean valuations and Archimedean valuations
		1.3. The Approximation Theorem and the Independence Theorem
		1.4. Valuations of the rational number field and of a rational function field
	§2. Complete fields
		2.1. Completions
		2.2. Remarks concerning normed spaces
		2.3. Extensions of a valuation of a complete field
	§3. Archimedean valuations
		3.1. The determination of complete Archimedean valuation fields
		3.2. Non-complete cases
	§4. Non-Archimedean valuations I
		4.1. Valuation rings, valuation ideals and residue class fields
		4.2. Hensel's Lemma
		4.3. Extensions of a valuation
		4.4. Discrete valuations
		4.5. The system of multiplicative representatives
	§5. Non-Archimedean valuations II
		5.1. Tensor products of vector spaces
		5.2. Compositions of fields
		5.3. Extensions of valuations
		5.4. Non-Archimedean valuations of an algebraic number field
		5.5. Valuations of an algebraic function field
	§6. Hilbert's theory
		6.1. Decomposition groups and inertia groups
		6.2. The decomposition of prime spots in intermediate fields
		6.3. Frobenius substitutions
		6.4. Ramification groups
		6.5. The ramification of a prime spot in intermediate fields
	§7. Discriminants and differents (local cases)
		7.1. Discriminants
		7.2. Differents
	§8. The differential of formal power series
CHAPTER III. Adele Rings and Idele Groups
	§1. Locally compact groups
		1.1. Haar measures
		1.2. Duality
		1.3. Fourier transformation
	§2. Locally compact rings
		2.1. The multiplicative topology
		2.2. Norms of invertible elements
		2.3. The topology of localJy compact fields
		2.4. The classification of locally compact fields
	§3. Local fields
		3.1. Local fields
		3.2. Self-duality
		3.3. The multiplicative group k_p^*
	§4. Adele and idele
		4.1. The restricted direct product
		4.2. Character groups
		4.3. Adele rings
		4.4. ldele groups
	§5. Extensions of the base field
		5.1. Isomorphisms of base fields
		5.2. Extensions of base fields
		5.3. Trace and norm
	§6. The structure of adele rings
		6.1. Expansion in partial fractions
		6.2. The structure of adele rings
		6.3. Self-duality
		6.4. Normal characters
	§7. The structure of idele groups
		7.1. The structure of idele groups
		7.2. The structure of idele class groups
CHAPTER IV. The Main Theorems of Class Field Theory
	§1. Cyclotomic fields
		1.1. General cyclotomic fields
		1.2. Cyclotomic fields over the rational number field
		1.3. Differents and discriminants of cyclotomic fields
	§2. Kummer fields
		2.1. Algebraic theory of Kummer fields
		2.2. Examples
	§3. Power residue symbols and Hilbert norm residue symbols
		3.1. Definitions of the symbols
		3.2. The self-duality of k_p^* / k_p^n (local cases)
		3.3. The global cases (self-duality of J_k / J_k^n)
	§4. Quadratic number fields and the Reciprocity Laws for quadratic residues
		4.1. Calculation of (a, b/2)
		4.2. Decomposition Law for quadratic number fields
		4.3. The Reciprocity Laws for quadratic residues
	§5. Artin-Schreier fields
		5.1. Artin-Schreier fields
		5.2. Artin-Schreier fields over k = F_q((t)) (Duality of k^* / k^P and k/ \mathscr{P}k )
		5.3. Artin-Schreier fields over an algebraic function field (The duality of J_k / J_k^p and R_k/ \mathscr{P}R_k )
	§6. The theory of infinite Galois extensions
		6.1. The Galois theory
		6.2. The maximum Abelian extensions and the character groups of their Galois groups
	§7. Main theorems of the class field theory
		7.1. Takagi groups and Artin correspondences
		7.2. Chevalley's formulation
		7.3. Ideal class groups, conductors
		7.4. The order of proofs
CHAPTER V. Proofs of the Main Theorems
	§1. Local cases
		1.1. Cohomology groups of local fields
		1.2. Invariants and canonical classes
		1.3. Norm residue symbol
		1.4. Relations with the Hilbert symbol
	§2. Proofs of the Conductor Theorems
		2.1. A theorem on the ramification groups
		2.2. The Conductor Theorems
	§3. The first inequality
		3.1. An extension of Herbrand's Lemma
		3.2. Calculation of Q(C_K)
	§4. The Second Inequality and the Existence Theorem
		4.1. Simplification of the problem
		4.2. Proof of the Second Inequality (case (a))
		4.3. Proof of the Existence Theorem (for case (a))
		4.4. Proof for case (b)
	§5. The Reciprocity Law
		5.1. Cyclotomic extensions and the Law of Reciprocity
		5.2. Idele class groups and their cohomology
		5.3. Summation formula of p-invariants and product formula of the norm residue symbols
	§6. Weil groups
		6.1. Weil groups and transfer
		6.2. The Weil groups; Local cases
		6.3. Algebraic function fields over finite fields and their Weil groups
		6.4. Weil groups of algebraic number fields
		6.5. Decomposition groups and Inertia groups
APPENDIX 1. Ideal Theory
	§1. Ideals in a Dedekind domain
		1.1. Dedekind domains
		1.2. Valuation theoretical characterization of a Dedekind domain
		1.3. Extensions of a Dedekind domain
		1.4. Extensions of ideals and their relative norms
	§2. Discriminants and differents (Global Cases)
		2.1. Differents
		2.2. Discriminants
		2.3. Miscellaneous theorems on differents and discriminants
		2.4. The absolute discriminant and Minkowski's Theorem
		2.5. An example
	§3. Artin-Whaples' Theory
APPENDIX 2. History of the Class Field Theory
	§1. From Euclid to Hilbert
		1.1. The birth of Number Theory and the accomplishment attained by Gauss
		1.2. Dirichlet and L-functions
		1.3. Kummer and Dedekind
		1.4. Complex multiplication
		1.5. Hilbert's 'Zahlbericht' and Hilbert's problems
		1.6. Hensel and p-adic numbers
	§2. Takagi and Artin's class field theory
		2.1. Weber's generalized ideal classes
		2.2. Takagi's class field theory I
		2.3. Takagi's class field theory II and Artin's Reciprocity Law
		2.4. Hasse's Bericht
	§3. The development of the theory after Takagi and Artin
		3.1. Simplification of the proofs by Artin, Herbrand, and Chevalley
		3.2. 'Arthmeticization' by Chevalley
		3.3. The class field theory and the algebra theory
		3.4. Hasse principle, idele and adele
		3.5. Algebraic function fields over finite constant fields, algebraic geometry
		3.6. The Weil groups, application of the cohomology theory and the class formation
		3.7. Ideal class groups, class numbers and some miscellaneous facts
Bibliography
Index