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Basic Abstract Algebra by S. K. Jain, S. R. Nagpaul, P. B. Bhattacharya

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CHAPTER 17

Galois theory

In this chapter we deal with the central results of Galois theory. The fundamental theorem on Galois theory establishes a one-to-one correspondence between the set of subfields of E, where E is a splitting field of a separable polynomial in F[x], and the set of subgroups of the group of F-automorphisms of E. This correspondence transforms certain problems about subfields of fields into more amenable problems about subgroups of groups. Among the applications, this serves as the basis of Galois’s criterion for solvability of an equation by radicals, as discussed in the next chapter, and provides a simple algebraic proof of the fundamental theorem of algebra.

1 Automorphism groups and fixed fields

Let E be an extension ...

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