Homework 6: Due May 2, 2007

1.

Find a basis for Q(2^1/4) and then a basis for Q(2^1/4)(2^1/4i).

2.

Let E=Q(2^1/4)(2^1/4i) and show it is a splitting field for p(x)= x⁴-2.

3.

Show that the group G_E of all field automorphisms $\sigma : E ->E$ that leave Q fixed is isomorphic to the dihedral group on the square. For each element of G_E, state explicitly what it does to the roots of p(x).

4.

For each subgroup G_B of G_E, find the intermediate field B that is left fixed by G_B. If G_B is a normal subgroup of G_E show that G_B is the splitting field of a separable polynomial by exhibiting the polynomial.

5.

Make a lattice diagram of the subgroups of G_E and label each with the corresponding intermediate field. Note that the containment relation between subfields is reversed to that of the containment relation of subgroups.