[[Galois theory]] uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The [[fundamental theorem of Galois theory]] provides a link between [[algebraic field extension]]s and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding [[Galois group]]. For example, ''S''<sub>5</sub>, the [[symmetric group]] in 5 elements, is not solvable which implies that the general [[quintic equation]] cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as [[class field theory]].
