Number Theory 2: Introduction to Class Field Theory

Cover
Title page
Contents
Preface to the English edition
Chapter 5. What is Class Field Theory?
	5.1. Examples of class field theoretic phenomena
		(a) Review
		(b) Decomposition of prime numbers in quadratic fields
		(c) Ramification and decomposition
		(d) Decomposition of prime numbers in fields other thanquadratic fields
		(e) Extension of algebraic number fields
		(f) Prime factors of polynomials
		(g) p = x^2 + 5y^2, p = x^2 + 6y^2, etc
	5.2. Cyclotomic fields and quadratic fields
		(a) The Galois group of a cyclotomic field
		(b) Decomposition of a prime number in a subfield of the cyclotomic field
		(c) Proofs of Theorems 5.4 and 5.7
		(d) Relations between cyclotomic fields and quadratic fields
		(e) Proofs of the relations between cyclotomic fields and quadratic fields
		(f) Proof of the quadratic reciprocity law `a la class field theory
	5.3. An outline of class field theory
		(a) Outline of the class field theory
		(b) p = x^2+5y^2, p = x^2+6y^2, . . . and class field theory
	Summary
	Exercises
Chapter 6. Local and Global Fields
	6.1. A curious analogy between numbers and functions
		(a) An analogy between integers and polynomials
		(b) Analogy between prime numbers and points
		(c) An analogy between p-adic numbers and Laurent series
		(d) An analogy between the place at infinity and the point at infinity
		(e) An analogy between algebraic number fields and algebraic function fields in one variable
		(f) Positive influences of the pursuit of analogies
	6.2. Places and local fields
		(a) Definition of places
		(b) Discrete valuation and discrete value ring
		(c) Completion
		(d) Global fields and local fields
		(e) Invariant measure and module
	6.3. Places and field extension
		(a) Decomposition of prime ideals in an extension field
		(b) Different and ramification
		(c) Decomposition of prime ideals and completion
		(d) Supplements on infinite places
	6.4. Adele rings and idele groups
		(a) Definitions of adele rings and idele groups
		(b) Relationship between the adele ring and the principal adeles
		(c) Relationship between the idele group and the principal ideles
		(d) Dirichlet’s unit theorem
		(e) Ideal class group as a quotient of the idele class group
		(f) Divisor class groups
		(g) Proofs of several results on discrete parts and compact quotients
		(h) Duality and denseness
		(i) A more precise formulation of ideal class group
	Summary
	Exercises
Chapter 7. ζ (II)
	7.1. The emergence of ζ
	7.2. Riemann ζ and Dirichlet L
		(a) Functional equation for Riemann ζ
		(b) Functional equation for Dirichlet L-functions
	7.3. Prime number theorems
		(a) Nonexistence of zeros
		(b) Riemann’s explicit formula
		(c) Prime number theorem
		(d) Riemann’s research on ζ
		(e) Dirichlet’s Prime Number Theorem
	7.4. The case of F_p[T]
	7.5. Dedekind ζ and Hecke L
	7.6. Generalization of the prime number theorem
	Summary
	Exercises
Chapter 8. Class Field Theory (II)
	8.1. The content of class field theory
		(a) “Easy-to-understand group” and Galois group
		(b) Maximal abelian extension
		(c) Theory of abelian extensions of finite fields
		(d) Main theorem in class field theory
		(e) What class field theory says — the local field case
		(f) What class field theory says — the global field case
		(g) What class field theory says—the algebraic numberfield case
		(h) What class field theory says—the function field case
		(i) Class field theory and Hecke characters
		(j) Construction problem of class fields
	8.2. Skew fields over a global or local field
		(a) The quaternion field of Hamilton
		(b) Quaternion fields and conics
		(c) Brauer group Br(k)
		(d) Brauer groups of global and local fields
		(e) Cyclic algebras
		(f) Relationship to class field theory
	8.3. Proof of the class field theory
	(a) Determination of the Brauer groups of local fields
	(b) Proof of local class field theory
	(c) Applications of ζ functions
	(d) Proof of Hasse’s reciprocity law
	(e) Proof of global class field theory (1)
	(f) Determination of the Brauer groups of algebraic number fields
	(g) Proof of the global class field theory (2)
	Summary
	Exercises
Appendix B. Galois Theory
	B.1. Galois theory
	B.2. Normal and separable extensions
	B.3. Norm and trance
	B.4. Finite fields
	B.5. Infinite Galois theory
Appendix C. Lights of Places
	C.1. Hensel’s lemma
	C.2. The Hasse principle
Answers to Questions
	Chapter 5
	Chapter 6
	Chapter 8
Answers to Exercises
	Chapter 5
	Chapter 6
	Chapter 7
	Chapter 8
Index
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