[Galois theory] Dimension and automorphisms

Hi,

I need some help with the following exercise.

Let . Show that and that has no automorphisms other than the identity (hint: has no root in and a single root in ).

I have no idea how I can prove and also no idea for the automorphism.

Can someone give me a hint?

Thanks in advance!

Re: [Galois theory] Dimension and automorphisms

is the degree of the minimal polynomial of , so the first one should be easy. Any automorphism on must fix the base field and send to one of the roots of its minimal polynomial. However, , so cannot be sent to a complex root, of which there are two. Hence can only be sent to itself, which means any automorphism is the identity.