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.