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Pure Picard-Vessiot extensions with generic properties

Author: Lourdes Juan
Journal: Proc. Amer. Math. Soc. 132 (2004), 2549-2556
MSC (2000): Primary 12H05; Secondary 12F12, 20G15
Published electronically: April 8, 2004
MathSciNet review: 2054779
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Abstract: Given a connected linear algebraic group $G$ over an algebraically closed field $C$ of characteristic 0, we construct a pure Picard-Vessiot extension for $G$, namely, a Picard-Vessiot extension $\mathcal E\supset \mathcal F$, with differential Galois group $G$, such that $\mathcal E$ and $\mathcal F$ are purely differentially transcendental over $C$. The differential field $\mathcal E$ is the quotient field of a $G$-stable proper differential subring $\mathcal R$ with the property that if $F$ is any differential field with field of constants $C$ and $E\supset F$ is a Picard-Vessiot extension with differential Galois group a connected subgroup $H$ of $G$, then there is a differential homomorphism $\phi:\mathcal R\rightarrow E$ such that $E$ is generated over $F$ as a differential field by $\phi(\mathcal R)$.

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Additional Information

Lourdes Juan
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409-1042

Received by editor(s): August 26, 2002
Received by editor(s) in revised form: June 2, 2003
Published electronically: April 8, 2004
Additional Notes: The author was supported in part by NSA grant No. MDA904-02-1-0084
Communicated by: Lance W. Small
Article copyright: © Copyright 2004 American Mathematical Society

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