The Theory of Algebraic Number Fields

Table of Contents

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   Book Overview
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   Chapter 1 Algebraic Numbers and Number Fields
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   Chapter 2 Ideals of Number Fields
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   Chapter 3 Congruences with Respect to Ideals
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   Chapter 4 The Discriminant of a Field and its Divisors
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   Chapter 5 Extension Fields
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   Chapter 6 Units of a Field
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   Chapter 7 Ideal Classes of a Field
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   Chapter 8 Reducible Forms of a Field
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    Chapter 9 Orders in a Field
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    Chapter 10 Prime Ideals of a Galois Number Field and its Subfields
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    Chapter 11 The Differents and Discriminants of a Galois Number Field and its Subfields
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    Chapter 12 Connexion Between the Arithmetic and Algebraic Properties of a Galois Number Field
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    Chapter 13 Composition of Number Fields
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    Chapter 14 The Prime Ideals of Degree 1 and the Class Concept
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    Chapter 15 Cyclic Extension Fields of Prime Degree
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    Chapter 16 Factorisation of Numbers in Quadratic Fields
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    Chapter 17 Genera in Quadratic Fields and Their Character Sets
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    Chapter 18 Existence of Genera in Quadratic Fields
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    Chapter 19 Determination of the Number of Ideal Classes of a Quadratic Field
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    Chapter 20 Orders and Modules of Quadratic Fields
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    Chapter 21 The Roots of Unity with Prime Number Exponent l and the Cyclotomic Field They Generate
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    Chapter 22 The Roots of Unity for a Composite Exponent m and the Cyclotomic Field They Generate
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    Chapter 23 Cyclotomic Fields as Abelian Fields
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    Chapter 24 The Root Numbers of the Cyclotomic Field of the l -th Roots of Unity
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    Chapter 25 The Reciprocity Law for l -th Power Residues Between a Rational Number and a Number in the Field of l -th Roots of Unity
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    Chapter 26 Determination of the Number of Ideal Classes in the Cyclotomic Field of the m -th Roots of Unity
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    Chapter 27 Applications of the Theory of Cyclotomic Fields to Quadratic Fields
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    Chapter 28 Factorisation of the Numbers of the Cyclotomic Field in a Kummer Field
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    Chapter 29 Norm Residues and Non-residues of a Kummer Field
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    Chapter 30 Existence of Infinitely Many Prime Ideals with Prescribed Power Characters in a Kummer Field
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    Chapter 31 Regular Cyclotomic Fields
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    Chapter 32 Ambig Ideal Classes and Genera in Regular Kummer Fields
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    Chapter 33 The l -th Power Reciprocity Law in Regular Cyclotomic Fields
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    Chapter 34 The Number of Genera in a Regular Kummer Field
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    Chapter 35 New Foundation of the Theory of Regular Kummer Fields
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    Chapter 36 The Diophantine Equation α m + β m + γ m = 0