Galois theory is an area of abstract algebra introduced by Evariste Galois, which 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, i.e. $$(\forall x\in F):\varphi(x)=x$$. 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 $$(\forall x\in K):\varphi(x)=x$$.