Editing Abel–Ruffini theorem (section)

===Galois correspondence===
The [[Galois correspondence]] establishes a [[one to one correspondence]] between the [[subextension]]s of a normal field extension <math>F/E</math> and the subgroups of the Galois group of the extension. This correspondence maps a field {{mvar|K}} such <math>E\subseteq K \subseteq F</math> to the [[Galois group]] <math>\operatorname{Gal}(F/K)</math> of the [[field automorphism|automorphisms]] of {{mvar|F}} that leave {{mvar|K}} fixed, and, conversely, maps a subgroup {{mvar|H}} of <math>\operatorname{Gal}(F/E)</math> to the field of the elements of {{mvar|F}} that are fixed by {{mvar|H}}.

The preceding section shows that an equation is solvable in terms of radicals if and only if the Galois group of its [[splitting field]] (the smallest field that contains all the roots) is [[solvable group|solvable]], that is, it contains a sequence of subgroups such that each is [[normal subgroup|normal]] in the preceding one, with a [[quotient group]] that is [[cyclic group|cyclic]]. (Solvable groups are commonly defined with [[abelian group|abelian]] instead of cyclic quotient groups, but the [[fundamental theorem of finite abelian groups]] shows that the two definitions are equivalent).

So, for proving the Abel–Ruffini theorem, it remains to show that the [[symmetric group]] <math>S_5</math> is not solvable, and that there are polynomials with symmetric Galois groups.