Topological Galois Theory by Askold Khovanskii

Author:Askold Khovanskii
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

This criterion is easy to deduce from Galois theory. Indeed, the monodromy group of an algebraic function is isomorphic to the Galois group of the field extension obtained from the field of all rational functions by adjoining all branches of the algebraic function.

In this section, we give a simple proof of the criterion that is independent of Galois theory and the other parts of this book (except that in Sect. 5.2.4, we use an uncomplicated linear-algebraic argument from Chap. 2). Sect. 5.2.1 provides an (almost obvious) verification of the fact that the monodromy groups of basic functions representable by radicals are solvable. In Sect. 5.2.2, we state the properties of solvable groups that we need. In Sect. 5.2.4, we prove that algebraic functions with solvable monodromy groups are representable by radicals.

Remark 5.7

The class of functions representable by radicals was defined in Sect. 1.​2 slightly differently. It is easy to see, however, that this definition is equivalent to the definition given above.

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