Centennial 2004

1904 - 2004

Dem Genius

Arnold Scholz

zum Gedächtnis

Portrait: Scholz

* 24. 12. 1904, Berlin     + 01. 02. 1942, Flensburg

Centennial 2004
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


The Connection between Cubic Fields and Dual Quadratic Fields

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

1. Connections between Cubic and Dual Quadratic Fields
2. Class Rank Configurations in SCHOLZ's Mirror Theorem


2-Stage Metabelian 3-Groups

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 Ka 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

0. Fundamental Facts
0.A. Generators and Relations
0.B. Transfers
1. Classical Scope
1.A. The Galois Group of the 2nd Hilbert 3-Class Field over a Real Quadratic Field
1.B. The Galois Group of the 2nd Hilbert 3-Class Field over a Complex Quadratic Field
1.C. Principalization Types
2. Modern Scope
2.A. The Galois Group of the 1st Hilbert 3-Class Field of the Absolute 3-Genus Field of a Cyclic Cubic Field
3. Top Recent Scope
3.A. The Galois Group of the 1st Hilbert 3-Class Field of the Relative 3-Genus Field of a Sextic S3 Field


3-Ring Class Fields over Quadratic Base Fields

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

1. Units and Ideal Classes
2. The Galois Group of the 1st Hilbert 3-Class Field of a Sextic S3 Field

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