## Tutorial 7 : Galois Groups of Quartics and Solvability by Radicals

1. Determine the Galois groups of the quartics:    and

2. Show that the resolvent cubic of is and

3. Show that the Galois group of an irreducible quartic in with exactly two real roots is either or

4. Let be a real root of an irreducible rational quartic whose resolvent cubic is irreducible. Show that is not constructible by ruler and compass. Can we construct the roots of the quartic by ruler and compass ?

5. Show that is irreducible over and

6. Find sufficient conditions on the integers and so that is a Galois extension of with cyclic Galois group of order

7. Let be an irreducible polynomial of prime degree Suppose that has exactly two non-real roots. Show that is not solvable by radicals over

8. Let be indeterminates and let be the elementary symmetric polynomials of Show that is not a radical extension of but is a radical extension of

9. Let be the Galois group of an irreducible quintic over Show that or if has an element of order