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Centennial 2004 |
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Kurz-Biographie von Arnold Scholz:
1904: geboren in Berlin-Charlottenburg 1911 - 1915: Vorschule 1915 - 1923: Kaiserin Augusta Gymnasium in Charlottenburg 1923 - 1928: Studium der Mathematik, Philosophie und Musikwissenschaft an der Universität Berlin 1927: ein Semester bei Ph. Furtwängler an der Universität Wien 1928: Promotio magna cum laude ("spondeo et polliceor") bei Issai Schur 1928 - 1930: Assistent an der Berliner Universität 1930 - 1935: Privat-Dozent in Freiburg (im Breisgau) 1935 - 1940: Lehrauftrag und Mitglied der Prüfungskommission in Kiel 1940: Kriegsdienst 1941: Mathematiklehrer an der Marine-Akademie Flensburg-Mürwick 1942: gestorben in Flensburg |
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The Connection between Cubic Fields and Dual Quadratic Fields |
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The deeper background of Cardano's Formula for the zeros of cubic polynomials:
--> 1925: W. H. E. Berwick classifies Non-Galois Cubic Fields with respect to the 3-class rank of the dual quadratic fields of the quadratic subfields of the Sextic Normal Fields only using elementary ideal theory and the influence of ideal cube generators, but unfortunately these rather deep results remain almost unknown --> 1933: Arnold Scholz establishes his famous Mirror Theorem concerning the connection between the 3-class ranks of dual quadratic fields by a simultaneous application of Class Field Theory and Kummer Theory |
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1. Connections between Cubic and Dual Quadratic Fields |
2. Class Rank Configurations in SCHOLZ's Mirror Theorem |
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2-Stage Metabelian 3-Groups |
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A research area with significant impact between 1925 and 1935:
--> Arising as relative automorphism groups G = Gal(N|K) of superfields N of some maximal abelian 3-extension K_{a} of a basefield K with the property that G' is abelian (G/G' is always abelian, anyway) --> 1927: Emil Artin establishes the Reciprocity Law of Class Field Theory and introduces the notion of transfer between commutator factor groups, inspired by works of Tchebotarev --> 1929: Ph. Furtwängler proves Hilbert's Principal Ideal Theorem using Artin's transfers --> 1934: Arnold Scholz and Olga Taussky investigate how ideal classes of a base field K with 3-class rank 2 become principal in the subfields of the Hilbert 3-class field of K, using Artin's transfers, and introducing the concept of capitulation of ideal classes |
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3-Ring Class Fields over Quadratic Base Fields |
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First applications of Class Field Theory exceeding the Cyclotomic Theory:
--> 1929: Helmut Hasse shows that elegant results on Non-Galois Cubic Fields can only be obtained by the arithmetic of their associated Cyclic Cubic Relative Extensions --> 1933: Arnold Scholz derives a complete classification of Non-Galois Cubic Fields according to their Unit Groups and Ideal Class Groups which is a coarse ancestor of the partition into Principal Factorization Types |
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1. Units and Ideal Classes |
2. The Galois Group of the 1^{st} Hilbert 3-Class Field of a Sextic S_{3} Field |
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