Cover; Title; Copyright; CONTENTS; Chapter I. ALGEBRAIC FIELDS; 1. Finite field. Norm, trace, discriminant; 2. Tower. Analysis of the field equation; 3. Simple extension; 4. Relative trace, norm and discriminant; 5. Removal of the hypothesis of separability; 6. The Galois case; 7. Consecutive extensions replaced by a single one; 8. Strictly finite field; 9. Adjunction of Indeterminate; Chapter II. THEORY OF DIVISIBILITY (KRONECKER, DEDEKIND); 1. Integers; 2. Our disbelief in Ideals; 3. The axioms; 4. Consequences; 5. Integrity in ϰ(x, y, .) over k(x, y, .); 6. Kronecker's theory.
7. The fundamental lemma8. A batch of simple propositions; 9. Relative Norm of a Divisor; 10. The Dedekind case; 11. Kronecker and Dedekind; Chapter III. LOCAL PRIMADIC ANALYSIS (KUMMER, HENSEL); 1. Quadratic number field; 2. Kummer's theory: decomposition; 3. Kummer's theory: discriminant; 4. Prime cyclotomic fields; 5. Program; 6. p-adic and y-adic numbers; 7. ϰ(y) and ϰ (J); 8. Discriminant; 9. Relative discriminant; 10. Hilbert's theory of Galois fields. Artin symbol; 11. Cyclotomlc field and quadratic law of reciprocity; 12. General cyclotomic fields; Chapter IV. ALGEBRAIC NUMBER FIELD.
1. Lattices (old-fashioned)2. Field basis and basis of an ideal; 3. Norm and number of residues; 4. Euler's function and Fermat's theorem; 5. A new viewpoint; 6. Minkowski's geometric principle; 7. A fundamental inequality and its consequences: existence of ramification ideals, classes of ideals; 8. The Dirichlet-Minkowski-Hasse-Chevalley construction of units; 9. The structure of the group of units; 10. Finite Abelian groups and their characters; 11. Asymptotic equi-distribution of ideals over their classes; 12. ζ-function and related Dirichlet series.
13. Prime numbers in residue classes modulo m14. ζ-function of quadratic fields, and their application; 15. Norm residues in quadratic fields; 16. General norm residue symbol and the theory of class fields; Amendments.
Ch. I. Algebraic Fields
Ch. II. Theory of Divisibility (Kronecker, Dedekind)
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