Extensions of the Field of Rational Numbers Whose Galois Group IS Isomorphic To A Group of Order Eight
Abstract
The Galois group of a certain type of field extension is a specific group associated with the field
extension. The study of field extensions and their relationship to the polynomials that give rise to
them via the Galois groups came to be called Galois Theory in honor of Everiste Galois who first
discovered them in 1830. In this project we find the extensions of ¤ whose Galois group is
isomorphic toC2 C2 C2 , 4 D , 4 2 C C and the quaternion group 8 Q . In each case we identify the
extension, then work out the subgroups of the Galois group and draw the lattice structure for the
subgroups. Also in each case, we work out by computation the fixed subfields of the extension
corresponding to the subgroups of the Galois group and draw their lattice structure.