Math 721

Spring 2001

Exam 2

1. Let be a field and let be an irreducible polynomial.

a) If is the splitting field of over and , show that acts transitively on the roots of

b) Show that if is irreducible of degree then is a transitive subgroup of

2. Find all transitive subgroups of

3. Let where is the real root of the polynomial

a) Show that is not Galois over

b) Let be the splitting field of the polynomial over Show that is Galois over and compute its Galois group.

c) Express all roots of the polynomial in terms of radicals.

4. Let be a finite field.

a) Show that for some prime

b) Show that any element in a finite field can be written as the sum of two squares.