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In section 2.3 we show Liouville’s First Theorem making use of algebraic teams actions. Of a generalization of the Second Theorem for linear differential equations of any order. A few words about our proof of the Second Liouville’s Theorem. All fundamental ideas of the proof are introduced within the book. Comments on them (principally no proofs are presented there). Solely some statements are lacking there (however all arguments needed for their proofs are offered in some form within the book). I feel that there are two statements which were not mentioned explicitly in the book: 1) a closed 1-kind with elementary integral whose attainable form was found by Liouville is locally invariant under the Galois group motion, assuming that the Galois group is related; 2) A category of closed 1-forms regionally invariant underneath a linked Lie group motion can be described explicitly. In case you look on a map, there is no actual separation between them.

Do you remember Christmas morning or your birthday, if you woke up and knew there have been presents only for you ready across the corner? I began to read it again after almost 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 deals with a topological approach to representability of algebraic functions by radicals and to the 13-th Hilbert problem. On an algebraic model of the 13-th Hilbert downside 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 method, which uses the speculation of complicated analytic functions and geometry, was very completely different from a formal algebraic approach. J.F.Ritt’s book accommodates principally all results needed for our version of the proof. I hope that this modern presentation will assist readers understand better the subject and J.F.Ritt’s book. The primary a part of the paper contains trendy proofs of The primary Theorem. In the second a part of the paper we present an overview of topological Galois idea which offers an alternative strategy to the issue of solvability of equations in finite phrases.