Let be a Field Extension of , denoted , and let be the set of Automorphisms of , that is, the set of Automorphisms of such that for every , so that is fixed. Then is a Group of transformations of , called the Galois group of .
The Galois group of consists of the Identity Element and Complex Conjugation. These functions both take a given Real to the same real.
See also Abhyankar's Conjecture, Finite Group, Group
Jacobson, N. Basic Algebra I, 2nd ed. New York: W. H. Freeman, p. 234, 1985.