The Theory of Algebraic Number Fields / by David Hilbert
| データ種別 | 電子ブック |
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| 出版者 | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer |
| 出版年 | 1998 |
| 本文言語 | 英語 |
| 大きさ | XXXVI, 351 p : online resource |
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| 内容注記 | 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 |
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| 一般注記 | This book is an English translation of Hilbert's Zahlbericht, the monumental report on the theory of algebraic number field which he composed for the German Mathematical Society. In this magisterial work Hilbert provides a unified account of the development of algebraic number theory up to the end of the nineteenth century. He greatly simplified Kummer's theory and laid the foundation for a general theory of abelian fields and class field theory. David Hilbert (1862-1943) made great contributions to many areas of mathematics - invariant theory, algebraic number theory, the foundations of geometry, integral equations, the foundations of mathematics and mathematical physics. He is remembered also for his lecture at the Paris International Congress of Mathematicians in 1900 where he presented a set of 23 problems "from the discussion of which an advancement of science may be expected" - his expectations have been amply fulfilled |
| 著者標目 | *Hilbert, David author SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:History LCSH:Number theory FREE:Mathematics FREE:Number Theory FREE:History of Mathematical Sciences |
| 分 類 | DC23:512.7 |
| 巻冊次 | ISBN:9783662035450 
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| ISBN | 9783662035450 |
| URL | http://dx.doi.org/10.1007/978-3-662-03545-0 |
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