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

(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

(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