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In part 2.Three we prove Liouville’s First Theorem making use of algebraic groups actions. Of a generalization of the Second Theorem for linear differential equations of any order. A couple of phrases about our proof of the Second Liouville’s Theorem. All major ideas of the proof are offered within the book. Feedback on them (basically no proofs are presented there). Only some statements are lacking there (but all arguments wanted for his or her proofs are introduced in some type in the book). I believe that there are two statements which weren’t mentioned explicitly within the book: 1) a closed 1-type with elementary integral whose possible form was found by Liouville is locally invariant underneath the Galois group motion, assuming that the Galois group is linked; 2) A class of closed 1-varieties locally invariant below a linked Lie group action might be described explicitly. In case you look on a map, there isn’t a actual separation between them.

Do you remember Christmas morning or your birthday, if you woke up and knew there were presents only for you waiting around the corner? I began to read it again after nearly a half century break. I simply began to work on topological obstructions to representability of algebraic capabilities by radicals. The first part of this half deals with a topological approach to representability of algebraic functions by radicals and to the 13-th Hilbert downside. On an algebraic version of the 13-th Hilbert drawback on representability of algebraic features of a number of advanced variables by composition of algebraic capabilities of fewer variety of variables. J.F.Ritt’s approach, which uses the idea of complex analytic features and geometry, was very totally different from a formal algebraic approach. J.F.Ritt’s book accommodates mainly all outcomes needed for our model of the proof. I hope that this trendy presentation will assist readers perceive better the subject and J.F.Ritt’s book. The primary a part of the paper comprises trendy proofs of The first Theorem. In the second a part of the paper we present an outline of topological Galois theory which offers an alternate approach to the problem of solvability of equations in finite terms.