The Theory of Algebraic Number Fields

David Hilbert (1862-1943) gilt als der vielleicht universellste Mathematiker des ausgehenden 19. und beginnenden 20. Jahrhunderts. Er hat auf zahlreichen Gebieten der Mathematik und der mathematischen Physik grundlegende neue Resultate vorgelegt und wesentliche Entwicklungen angebahnt.

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.