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The Theory of Algebraic Number Fields - David Hilbert

The Theory of Algebraic Number Fields

By: David Hilbert, Franz Lemmermeyer (Introduction by), Norbert Schappacher (Introduction by), Rene Schoof (Introduction by), I.T. Adamson (Translator)


Published: 20th August 1998
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A translation of Hilberts "Theorie der algebraischen Zahlkoerper" best known as the "Zahlbericht", first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher.

Translator's Preface
Hilbert's Preface
Introduction to the English Edition
The Theory of General Number Fields
Algebraic Numbers and Number Fieldsp. 3
Ideals of Number Fieldsp. 9
Congruences with Respect to Idealsp. 17
The Discriminant of a Field and its Divisorsp. 25
Extension Fieldsp. 33
Units of a Fieldp. 41
Ideal Classes of a Fieldp. 53
Reducible Forms of a Fieldp. 65
Orders in a Fieldp. 67
Galois Number Fields
Prime Ideals of a Galois Number Field and its Subfieldsp. 79
The Differents and Discriminants of a Galois Number Field and its Subfieldsp. 89
Connexion Between the Arithmetic and Algebraic Properties of a Galois Number Fieldp. 93
Composition of Number Fieldsp. 97
The Prime Ideals of Degree 1 and the Class Conceptp. 101
Cyclic Extension Fields of Prime Degreep. 105
Quadratic Number Fields
Factorisation of Numbers in Quadratic Fieldsp. 115
Genera in Quadratic Fields and Their Character Setsp. 121
Existence of Genera in Quadratic Fieldsp. 133
Determination of the Number of Ideal Classes of a Quadratic Fieldp. 149
Orders and Modules of Quadratic Fieldsp. 155
Cyclotomic Fields
The Roots of Unity with Prime Number Exponent l and the Cyclotomic Field They Generatep. 161
The Roots of Unity for a Composite Exponent m and the Cyclotomic Field They Generatep. 167
Cyclotomic Fields as Abelian Fieldsp. 175
The Root Numbers of the Cyclotomic Field of the l-th Roots of Unityp. 187
The Reciprocity Law for l-th Power Residues Between a Rational Number and a Number in the Field of l-th Roots of Unityp. 199
Determination of the Number of Ideal Classes in the Cyclotomic Field of the m-th Roots of Unityp. 207
Applications of the Theory of Cyclotomic Fields to Quadratic Fieldsp. 217
Kummer Number Fields
Factorisation of the Numbers of the Cyclotomic Field in a Kummer Fieldp. 225
Norm Residues and Non-residues of a Kummer Fieldp. 233
Existence of Infinitely Many Prime Ideals with Prescribed Power Characters in a Kummer Fieldp. 253
Regular Cyclotomic Fieldsp. 257
Ambig Ideal Classes and Genera in Regular Kummer Fieldsp. 269
The l-th Power Reciprocity Law in Regular Cyclotomic Fieldsp. 289
The Number of Genera in a Regular Kummer Fieldp. 305
New Foundation of the Theory of Regular Kummer Fieldsp. 313
The Diophantine Equation [alpha][superscript m] + [beta][superscript m] + [gamma][superscript m] = 0p. 327
Referencesp. 335
List of Theorems and Lemmasp. 345
Indexp. 347
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783540627791
ISBN-10: 3540627790
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 351
Published: 20th August 1998
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 24.13 x 16.51  x 2.54
Weight (kg): 0.7