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This is an English translation of "The Problem of Resolvents and CriticalManifolds" by Tschebotarow/Chebotarev. In this article, Chebotarev explains hiswork on resolvent problems using critical manifolds. The current ideas ofresolvent degree and essential dimension arose out of the resolvent problemsChebotarev addresses here.  Original abstract by Chebotarev: "This paper is devoted to the study of theproblem of resolvents, i.e., the problem of finding the resolvent by a givenequation, whose coefficients depend on several independent parameters, thenumber of parameters in the coefficients being as small as possible. The authorconnects the problem to the study of critical manifolds in the space ofequation parameters."

Translation of Tschebotarow's "The Problem of Resolvents and Critical Manifolds"