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Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use (galois-connections).

Galois theory is an area of abstract algebra introduced by Evariste Galois. The key point is that it provides a connection between field theory and group theory. Given a field $F$ and an extension $F$ of $E$ with certain properties (a type of extension called Galois extension), let $\operatorname{Gal}(E/F)$ be the group of automorphisms $\varphi$ of $E$ which leave $F$ fixed, that is, such that $(\forall x\in F):\varphi(x)=x$. The fundamental theorem of Galois theory asserts that there is a one-to one correspondence between subfields of $E$ which are extensions of $F$ and subgroups of $\operatorname{Gal}(E/F)$:

  • if $H$ is a subgroup of $\operatorname{Gal}(E/F)$, then the set of those $x\in E$ such that $(\forall\varphi\in H):\varphi(x)=x$ is a subfield of $E$ which is an extension of $F$;
  • to each subfield $K$ of $E$ which is an extension of $F$, one can associate the subgroup of $\operatorname{Gal}(E/F)$ whose elements are those $\varphi\in\operatorname{Gal}(E/F)$ such that $(\forall x\in K):\varphi(x)=x$.
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