Galois theory

In mathematics, Galois theory is a branch of abstract algebra.

The inverse Galois problem

See main article: inverse Galois problem

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It is easy to construct field extensions with any given finite group as Galois group. That is, all finite groups do occur as Galois groups.

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For that, choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates {xα}, one for each element α of G, and adjoin them to K to get the field F = K({xα}). Contained within F is the field L of symmetric rational functions in the {xα}. The Galois group of F/L is S, by a basic result of Emil Artin. G acts on F by restriction of action of S. If the fixed field of this action is M, then, by the fundamental theorem of Galois theory, the Galois group of F/M is G.

Related Topics:
Cayley's theorem - Symmetric group - Rational function - Emil Artin - Fundamental theorem of Galois theory

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It is an open problem (in general) how to construct field extensions of a fixed ground field with a given finite group as Galois group.

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