The Theory of Algebraic Number Fields

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