Number Theory

Number Theory

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From the reviews: "...a fine book [...] When it appeared in 1949 it was a pioneer. Now there are plenty of competing accounts. But Hasse has something extra to offer.[...] Hasse proved that miracles do happen in his five beautiful papers on quadratic forms of 1923-1924. [...]It is trite but true: Every number-theorist should have this book on his or her shelf." --Irving Kaplansky in Bulletin of the American Mathematical Society, 1981
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Product details

  • Paperback | 640 pages
  • 170 x 244 x 36.58mm | 1,160g
  • Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Berlin, Germany
  • English
  • Reprint of the 1st ed
  • XVII, 640 p.
  • 354042749X
  • 9783540427490
  • 1,764,730

Table of contents

Part I. The Foundations of Arthmetic in the Rational Number Field:
Chapter 1 Prime Decomposition
Chapter 2 Divisibility
Chapter 3 Congruences
Chapter 4 The Structure of the Residue Class Ring mod m and the Reduces Residue Class Group mod m.
Chapter 5 Quadratic Residues

Part II. The Theory of Valued Fields
Chapter 6 The Fundamental Concepts Regarding Valuations
Chapter 7 Arithmetic in a Discrete Valued Field
Chapter 8 The Completition of a Valued Field
Chapter 9 The Completition of a Discrete Valued Field. The p-adic Number Fields
Chapter 10 The Isomorphism Types of Complete Discrete Valued Fields with Perfect Residue Class Field
Chapter 11 Prolongation of the Discrete Valuation to a Purely Transcendental Extension
Chapter 12 Prolongation of the Valuation of a Complete Field to a Finite Algebraic Extension
Chapter 13 The Isomorphism Types of Complete Archidemean Valued Fields
Chapter 14 The Structure of a Finite-Algebraic Extension of a Complete Discrete Valued Field
Chapter 15 The Structure of the Multiplicative Group of a Complete Discrete Valued Field with Perfect Residue Class Fields of Prime Characteristic
Chapter 16 The Tamely Ramified Extension Types of a Complete Discrete Valued Fields with Finite Residue Class Field of Characteristic p
Chapter 17 The Exponential Function, the Logarithm, and Powers in a Complete Non-Archimedean Valued Field of Characteristic 0
Chapter 18 Prolongation of the Valuation of a Non-Complete Field to a Finite-Algebraic Extension

Part III. The Foundations of Arithmetic Algebraic Number Fields
Chapter 19 Relations Between the Complete System of Valuations and the Arithmetic of the Rational Number Field
Chapter 20 Prolongation of the Complete System of Valuations to a Finite-Algebraic Extension
Chapter 21 The Prime Spots of an Algebraic Number Field and their Completions
Chapter 22 Decomposition into Prime Divisors, Integrality, and Divisibility
Chapter 23 Congruences
Chapter 24 The Multiples of a Divisor
Chapter 25 Differents and Discriminants
Chapter 26 Quadratic Number Fields
Chapter 27 Cyclotomic Fields
Chapter 28 Units
Chapter 29 The Class Number
Chapter 30 Approximation Theorems and Estimates of the Discriminant
Index of Names
Subject Index
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About Helmut Hasse

Biography of Helmut Hasse (1898-1979)

Born on August 25, 1898 in Kassel, Germany, Helmut Hasse studied at the University of Goettingen after WWI. Of his teachers there including Landau, Hilbert and Ehmy Noether, Hecke influenced him most. In 1820, Hasse went to Marburg, and under the direction of Kurt Hensel, discovered what is now known as the Hasse principle, or "local-global" principle, in algebraic number theory. He held further positions at the universities in Kiel and Hall prior to 1933. With the troubles of 1933, Hermann Weyl, who had succeeded Hilbert in the foremost chair or mathematics in Germany, resigned and Helmut Hasse was appointed in this place. The following year, Hasse became director of the Mathematical Institute at Goettingen. From 1939 to 1945, Hasse worked in Berlin for the navy on problems in ballistics. He returned to Goettingen but was soon dismissed by the British occupation forces. In 1946 he took a research position at the Berlin Academy. Thereafter, he held positions at the Humboldt University in East Berlin, and, from 1950 until retirement in 1966, at the University of Hamburg.

At Halle, Hasse obtained fundamental results on the structure of central simple algebras over local fields. In Marburg, he did joint work with Brauer and Emmy Noether on simple algebras, also on elliptic curves and topological fields. In particular, he proved the analogon of the Riemann Hypothesis for zeta functions of elliptic curves. Both of Hasse?s famous books UEber die Klassenzahl abelscher Zahlkoerper und Zahlentheorie appeared during his years in Berlin.
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