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E-raamat: Theory of Algebraic Number Fields

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 14-Mar-2013
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662035450
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Theory of Algebraic Number FieldsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 14-Mar-2013
  • Kirjastus: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Keel: eng
  • ISBN-13: 9783662035450
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Constance Reid, in Chapter VII of her book Hilbert, tells the story of the writing of the Zahlbericht, as his report entitled Die Theorie der algebra is­ chen Zahlkorper has always been known. At its annual meeting in 1893 the Deutsche Mathematiker-Vereinigung (the German Mathematical Society) invited Hilbert and Minkowski to prepare a report on the current state of affairs in the theory of numbers, to be completed in two years. The two mathematicians agreed that Minkowski should write about rational number theory and Hilbert about algebraic number theory. Although Hilbert had almost completed his share of the report by the beginning of 1896 Minkowski had made much less progress and it was agreed that he should withdraw from his part of the project. Shortly afterwards Hilbert finished writing his report on algebraic number fields and the manuscript, carefully copied by his wife, was sent to the printers. The proofs were read by Minkowski, aided in part by Hurwitz, slowly and carefully, with close attention to the mathematical exposition as well as to the type-setting; at Minkowski's insistence Hilbert included a note of thanks to his wife. As Constance Reid writes, "The report on algebraic number fields exceeded in every way the expectation of the members of the Mathemati­ cal Society. They had asked for a summary of the current state of affairs in the theory. They received a masterpiece, which simply and clearly fitted all the difficult developments of recent times into an elegantly integrated theory.
Translators Preface V(2) Hilberts Preface VII(16) Introduction to the English Edition XXIII Franz Lemmermeyer Norbert Schappacher
1. The Report XXIII(2)
2. Later Criticism XXV(3)
3. Kummers Theory XXVIII(4)
4. A Few Noteworthy Details XXXII Part I. The Theory of General Number Fields 3(76)
1. Algebraic Numbers and Number Fields 3(6)
1. Number Fields and Their Conjugates 3(1)
2. Algebraic Integers 4(1)
3. Norm, Different and Discriminant of a Number. Basis of a Number Field 5(4)
2. Ideals of Number Fields 9(8)
4. Multiplication and Divisibility of Ideals. Prime Ideals 9(2)
5. Unique Factorisation of an Ideal into Prime Ideals 11(3)
6. Forms of Number Fields and Their Contents 14(3)
3. Congruences with Respect to Ideals 17(8)
7. The Norm of an Ideal and Its Properties 17(3)
8. Fermats Theorem in Ideal Theory. The Function Phi(a) 20(2)
9. Primitive Roots for a Prime Ideal 22(3)
4. The Discriminant of a Field and its Divisors 25(8)
10. Theorem on the Divisors of the Discriminant. Lemma on Integral Functions 25(3)
11. Factorisation and Discriminant of the Fundamental Equation 28(2)
12. Elements and Different of a Field. Proof of the Theorem on the Divisors of the Discriminant of a Field 30(1)
13. Determination of Prime Ideals. Constant Numerical Factors of the Rational Unit Form U 31(2)
5. Extension Fields 33(8)
14. Relative Norms, Differents and Discriminants 33(2)
15. Properties of the Relative Different and Discriminant 35(3)
16. Decomposition of an Element of a Field k in an Extension K. Theorem on the Different of the Extension K 38(3)
6. Units of a Field 41(12)
17. Existence of Conjugates with Absolute Values Satisfying Certain Inequalities 41(2)
18. Absolute Value of the Field Discriminant 43(2)
19. Theorem on the Existence of Units 45(4)
20. Proof of the Theorem on the Existence of Units 49(2)
21. Fundamental Sets of Units. Regulator of a Field. Independent Sets of Units 51(2)
7. Ideal Classes of a Field 53(12)
22. Ideal Classes. Finiteness of the Class Number 53(1)
23. Applications of the Theorem on the Finiteness of the Class Number 54(2)
24. The Set of Ideal Classes. Strict Form of the Class Concept 56(1)
25. A Lemma on the Asymptotic Value of the Number of All Principal Ideals Divisible by a Given Ideal 56(4)
26. Determination of the Class Number by the Residue of the Function Zeta(s) at s = 1 60(2)
27. Alternative Infinite Expansions of the Function Zeta(s) 62(1)
28. Composition of Ideal Classes of a Field 62(2)
29. Characters of Ideal Classes. Generalisation of the Function Zeta(s) 64(1)
8. Reducible Forms of a Field 65(2)
30. Reducible Forms. Form Classes and Their Composition 65(2)
9. Orders in a Field 67(12)
31. Orders. Order Ideals and Their Most Important Properties 67(2)
32. Order Determined by an Integer. Theorem on the Different of an Integer of a Field 69(3)
33. Regular Order Ideals and Their Divisibility Laws 72(1)
34. Units of an Order. Order Ideal Classes 73(1)
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