Khovanskij, A. G. On the solvability and unsolvability of equations in explicit form. (English. Russian original) Zbl 1069.34133 Russ. Math. Surv. 59, No. 4, 661-736 (2004); translation from Usp. Mat. Nauk 59, No. 4, 69-146 (2004). The survey is devoted to the solvability and unsolvability of equations in explicit form. The classical theory of Abel, Liouville, Galois, Picard, Vessiot, Kolchin and others is described and discussed. For example, the one-dimensional topological version of Galois theory and the solvability of linear differential equations by quadratures and the Picard-Vessiot theory is discussed in detail. The survey is an excellent summary of the actual existing methods. Reviewer: J. Saurer (Regensburg) Cited in 1 ReviewCited in 3 Documents MSC: 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 34A05 Explicit solutions, first integrals of ordinary differential equations 12F10 Separable extensions, Galois theory 12H99 Differential and difference algebra 34A30 Linear ordinary differential equations and systems 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations Keywords:solvability of equations; Liouville theory; Galois theory; Picard-Vessiot theory; quadratures PDFBibTeX XMLCite \textit{A. G. Khovanskij}, Russ. Math. Surv. 59, No. 4, 661--736 (2004; Zbl 1069.34133); translation from Usp. Mat. Nauk 59, No. 4, 69--146 (2004) Full Text: DOI