This is an English translation of Felix Kleins classical paper Uber die Auflosung der allgemeinen Gleichungen funften und sechsten Grades (Auszug aus einem Schreiben an Herrn K. Hensel) from 1905 and is put in modern notation. The original work first appeared in the Journal for Pure and Applied Mathematics (Volume 129) and then was reprinted in Mathematische Annalen (Volume 61, Issue 1). Kleins work (including his Lectures on the Icosahedron and the Solution of Equations of Fifth Degree) lies at the heart of the 19th and 20th work on solving generic polynomials. In this paper, Klein summarizes his approach to solving the generic quintic and sextic polynomials. He also lays the foundation for the modern framework of resolvent degree.
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