Galois Group

In Galois theory, a branch of abstract algebra, the Galois group of a field extension is a symmetry group that characterizes how the extension relates to its base field, with each element of the group a transformation of the extension that leaves every element of the base field fixed. The fundamental theorem of Galois theory connects fields and groups in this way, letting group-theoretic tools be brought to bear on problems in field theory, such as describing the solutions of quintic polynomials; the whole subject of studying field extensions through their Galois groups is named Galois theory, in honor of Evariste Galois, who first discovered them.