Mathematics 894 Problem Set 11

Do not hand in: Solutions will (hopefully) be provided. Let K be the splitting field of x^{4} + 2 over ℚ.

1. Prove that K = ℚ(\sqrt[4]{2}, i). (This way of representing K seems simpler than the way provided in the solutions to Problem Set 9.)

2. Prove that [K : ℚ] = 8.

3. Find G = Gal(K/ℚ), representing G by the action of its elements on \{\sqrt[4]{2}, i\} and as a permutation group of the roots of x^{4} + 2.

4. Give the lattice of subgroups of G and the corresponding subfields of K as is done for several examples in section 14.2 of Dummit and Foote.

5. Which subfields of K are Galois over ℚ?

6. Find a primitive element of K over ℚ. Verify that your candidate is primitive by considering how G acts on it.