Irreducibility of the linear differential equation attached to Painlevé's first equation (Q690147)

The following theorem is proved: Let \(K\) be a differential field of characteristic 0 (with derivation \(a\mapsto a'\)) whose field of constants is algebraically closed in a universal extension of \(K\) and which contains an element \(x\) such that \(x'=1\); let \(y\) be the general solution of Painlevé's first equation \(y''= 6y^ 2+ x\). Then the Picard-Vessiot group of the differential field extension \(K\langle y\rangle\) of \(K\) is irreducible. -- See also the author's paper [Nagoya Math. J. 109, 63-67 (1988; Zbl 0613.34030)].