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In section 2.3 we show 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 number of words about our proof of the Second Liouville’s Theorem. All principal concepts of the proof are presented in the book. Feedback on them (basically no proofs are offered there). Solely some statements are lacking there (but all arguments wanted for their proofs are presented in some form within the book). I believe that there are two statements which were not talked about explicitly within the book: 1) a closed 1-type with elementary integral whose doable form was found by Liouville is regionally invariant under the Galois group motion, assuming that the Galois group is linked; 2) A class of closed 1-types locally invariant underneath a connected Lie group action might be described explicitly. When you look on a map, there isn’t a actual separation between them.

Do you remember Christmas morning or your birthday, whenever you woke up and knew there were presents only for you waiting across the nook? I began to read it again after nearly a half century break. I just started to work on topological obstructions to representability of algebraic features by radicals. The primary section of this half offers with a topological strategy to representability of algebraic functions by radicals and to the 13-th Hilbert drawback. On an algebraic model of the 13-th Hilbert drawback on representability of algebraic capabilities of a number of advanced variables by composition of algebraic capabilities of fewer variety of variables. J.F.Ritt’s approach, which uses the theory of complicated analytic capabilities and geometry, was very completely different from a formal algebraic method. J.F.Ritt’s book incorporates mainly all outcomes wanted for our model of the proof. I hope that this trendy presentation will help readers perceive better the topic and J.F.Ritt’s book. The first part of the paper contains modern proofs of The primary Theorem. Within the second a part of the paper we current an overview of topological Galois idea which provides an alternative approach to the problem of solvability of equations in finite terms.