Author | Hilbert, David. author |
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Title | The Theory of Algebraic Number Fields [electronic resource] / by David Hilbert |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1998 |
Connect to | http://dx.doi.org/10.1007/978-3-662-03545-0 |
Descript | XXXVI, 351 p. online resource |
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