Number Theory

Part I. The Foundations of Arthmetic in the Rational Number Field:Chapter 1 Prime DecompositionChapter 2 DivisibilityChapter 3 CongruencesChapter 4 The Structure of the Residue Class Ring mod m and the Reduces Residue Class Group mod m.Chapter 5 Quadratic ResiduesPart II. The Theory of Valued FieldsChapter 6 The Fundamental Concepts Regarding ValuationsChapter 7 Arithmetic in a Discrete Valued FieldChapter 8 The Completition of a Valued FieldChapter 9 The Completition of a Discrete Valued Field. The p-adic Number FieldsChapter 10 The Isomorphism Types of Complete Discrete Valued Fields with Perfect Residue Class FieldChapter 11 Prolongation of the Discrete Valuation to a Purely Transcendental ExtensionChapter 12 Prolongation of the Valuation of a Complete Field to a Finite Algebraic ExtensionChapter 13 The Isomorphism Types of Complete Archidemean Valued FieldsChapter 14 The Structure of a Finite-Algebraic Extension of a Complete Discrete Valued FieldChapter 15 The Structure of the Multiplicative Group of a Complete Discrete Valued Field with Perfect Residue Class Fields of Prime CharacteristicChapter 16 The Tamely Ramified Extension Types of a Complete Discrete Valued Fields with Finite Residue Class Field of Characteristic pChapter 17 The Exponential Function, the Logarithm, and Powers in a Complete Non-Archimedean Valued Field of Characteristic 0Chapter 18 Prolongation of the Valuation of a Non-Complete Field to a Finite-Algebraic ExtensionPart III. The Foundations of Arithmetic Algebraic Number FieldsChapter 19 Relations Between the Complete System of Valuations and the Arithmetic of the Rational Number FieldChapter 20 Prolongation of the Complete System of Valuations to a Finite-Algebraic ExtensionChapter 21 The Prime Spots of an Algebraic Number Field and their CompletionsChapter 22 Decomposition into Prime Divisors, Integrality, and DivisibilityChapter 23 CongruencesChapter 24 The Multiples of a DivisorChapter 25 Differents and DiscriminantsChapter26

Biography of Helmut Hasse (1898-1979)

Born on August 25, 1898 in Kassel, Germany, Helmut Hasse studied at the University of Göttingen 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 Göttingen. From 1939 to 1945, Hasse worked in Berlin for the navy on problems in ballistics. He returned to Göttingen 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 Über die Klassenzahl abelscher Zahlkörper und Zahlentheorie appeared during his years in Berlin.