Solvability of Algebraic Equations by Radicals and Galois Theory
Is a given algebraic equation solvable by radicals? Can one solve a given algebraic equation of degree n using solutions of auxiliary algebraic equations of smaller degree and radicals? In this chapter, we discuss how Galois theory answers these questions
- PDF / 431,248 Bytes
- 37 Pages / 439.36 x 666.15 pts Page_size
- 108 Downloads / 341 Views
Solvability of Algebraic Equations by Radicals and Galois Theory
Is a given algebraic equation solvable by radicals? Can one solve a given algebraic equation of degree n using solutions of auxiliary algebraic equations of smaller degree and radicals? In this chapter, we discuss how Galois theory answers these questions (at least in principle). The questions we have posed are purely algebraic by nature and can be stated over any field K. We will assume in this chapter that the field K has characteristic zero. This case is slightly simpler than the case of general characteristic, and we are mostly interested in functional differential fields, which contain all complex constants. Other interesting examples of fields to which the results of this chapter are applicable are subfields of the field of complex numbers (in particular, the field of all rational numbers). The “permissive” part of Galois theory (see Sect. 2.1) that allows us to solve equations by radicals is very simple. It depends neither on the fundamental theorem of Galois theory nor on the theory of fields, and it essentially belongs to linear algebra. Only these linear-algebraic considerations are used in the topological version of Galois theory in relationship to the question of representability of algebraic functions by radicals. However, a sufficient condition for solvability of an equation by solving auxiliary equations of smaller degree and taking radicals depends not only on linear algebra, but also on the fundamental theorem of Galois theory. This is one of the reasons that we give a complete proof of that theorem. The well-known properties of solvable groups and the symmetric groups S.k/ are used without proof. In Sect. 2.9.1, we prove a considerably less well known characteristic property of the subgroups of S.k/. These facts from group theory are applied in usual Galois theory as well as in its differential and topological versions. We often need to extend the field under consideration by adjoining one or several roots of an algebraic equation. For functional differential fields, this construction is simple and was already described in Sect. 1.5. For subfields of the field of complex numbers, the construction of such extensions is obvious. Since we are
© Springer-Verlag Berlin Heidelberg 2014 A. Khovanskii, Topological Galois Theory, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-38871-2__2
47
48
2 Solvability of Algebraic Equations by Radicals and Galois Theory
mostly interested in fields of these two types, we will use such extensions below without giving details on how to construct them. Several words are in order on the organization of the material. In Sects. 2.1– 2.4, we consider a field P on which a finite group G acts by field automorphisms. Elements of the field P fixed under the action of G form a subfield K P called the invariant subfield of G. In Sect. 2.1, we show that if the group G is solvable, then the elements of the field P are representable by radicals through the elements of the invariant subfield K of G.
Data Loading...
