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AuthorHilbert, David. author
TitleThe Theory of Algebraic Number Fields [electronic resource] / by David Hilbert
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1998
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Descript XXXVI, 351 p. online resource


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


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

Mathematics History Number theory Mathematics Number Theory History of Mathematical Sciences


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