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On the Inverse Problem in Differential Galois Theory

Hartmann, Julia

German Title: Zum inversen Problem der Differentialgaloistheorie

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Abstract

Differential Galois theory generalizes the usual Galois theory for polynomials to differential equations. There is the notion of a splitting field (Picard-Vessiot extension) of a differential equation, and the differential Galois group is the group of automorphisms of this extension which fix the base field and commute with the derivation. Differential Galois groups are linear algebraic groups over the field of constants of the base field. In analogy to the classical situation, one considers the inverse problem: Which linear algebraic groups occur as differential Galois groups over a given differential field? The main result of this thesis is the following theorem: Every linear algebraic group defined over the algebraically closed field K occurs as the differential Galois group of some Picard-Vessiot extension of K(t) with derivation d/dt.

Translation of abstract (German)

Die Differentialgaloistheorie stellt eine Verallgemeinerung der gewoehnlichen Galoistheorie auf Differentialgleichungen dar. Dem Zerfaellungskoerper entspricht bei Differentialgleichungen die Picard-Vessiot-Erweiterung, und die Differentialgaloisgruppe ist die Gruppe derjenigen Automorphismen dieser Erweiterung, die den Grundkoerper festlassen und mit der gegebenen Derivation vertraeglich sind. Die Differentialgaloisgruppen sind lineare algebraische Gruppen, definiert ueber dem Konstantenkoerper des Grundkoerpers. Analog zur klassischen Situation betrachtet man das inverse Problem: Welche linearen algebraischen Gruppen treten als Differentialgaloisgruppen ueber einem gegebenen Differentialkoerper auf? Das Hauptresultat dieser Arbeit ist der folgende Satz: Jede ueber dem algebraisch abgeschlossenen Koerper K definierte lineare algebraische Gruppe ist Differentialgaloisgruppe einer Picard-Vessiot-Erweiterung von K(t) mit Derivation d/dt.

Item Type: Dissertation
Supervisor: Matzat, Prof. Dr. B. Heinrich
Date of thesis defense: 21 October 2002
Date Deposited: 19 Dec 2002 08:42
Date: 2002
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Mathematics
Subjects: 510 Mathematics
Controlled Keywords: Galois-Theorie, Umkehrproblem <Galois-Theorie>, Differentialalgebra
Uncontrolled Keywords: DifferentialgaloistheorieDifferential Galois theory
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