The differential Galois group of the rational function field

Michael Wibmer, Julia Hartmann, David Harbater, Annette Bachmayr

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Article number107605
JournalAdvances in Mathematics
Publication statusPublished - 2021


  • Differential algebra
  • Embedding problems
  • Inverse differential Galois problem
  • Linear algebraic groups
  • Picard-Vessiot theory
  • Proalgebraic groups

ASJC Scopus subject areas

  • Mathematics(all)


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