Abstract
We determine the absolute differential Galois group of the field C(x) of rational functions: It is the free proalgebraic group on a set of cardinality |C|. This solves a longstanding open problem posed by B.H. Matzat. For the proof we develop a new characterization of free proalgebraic groups in terms of split embedding problems, and we use patching techniques in order to solve a very general class of differential embedding problems. Our result about C(x) also applies to rational function fields over more general fields of coefficients.
Original language | English |
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Article number | 107605 |
Journal | Advances in Mathematics |
Volume | 381 |
DOIs | |
Publication status | Published - 2021 |
Keywords
- Differential algebra
- Embedding problems
- Inverse differential Galois problem
- Linear algebraic groups
- Picard-Vessiot theory
- Proalgebraic groups
ASJC Scopus subject areas
- General Mathematics