## Tutorial 5 : Fundamental Theorem of Galois Theory

1. Let be an irreducible quartic with Galois group over Let be a root of Show that there is no field properly contained in Is a Galois extension ?

2. Consider the biquadratic extension of Find the intermediate subfields of and match them with the subgroups of the Galois group

3. Let where is a transcendental over Let Define the -automorphisms and of by the equations and Show that
and
Show that the group of automorphisms generated by and has order and

4. Consider the polynomial

(a)
Show that the roots of the quartic are
and
(b) Show : and
(c)
Show that and are Galois over
(d)
Show that is the Klein -group. Determine the automorphisms in             this group.
(e) Show that the Galois group of over is the dihedral of order

5. Let denote the rational function field in the indeterminate over Let and be the automorphism that substitutes by Put Show that

6. Suppose that the Galois group of a field extension is the Klein -group Show that is biquadratic.

7. Let where is a root of in Show that Determine

8. Let be variables. Let and Show that is a Galois extension of of degree Find