Splitting Fields and Galois Groups

Hi,

I've been set a question and think I've managed about half of it but stuck on some other bits..

Show that is irreducible over . Let be given as a root of . Show that is also a root of . Hence, find a splitting field for over . Identity the Galois group . (You must justify any claim that a particular map is an isomorphism. Be sure to state carefully

any facts that you assume in doing this.)

first part is easy just reducing by to get \ which has no zeros in and thus irreducible over and hence over also.

I think I can identify the Galois group using Prop 22.4 which gives , a perfect square thus the Galois Group is

I can't think how to show that is also a root of . For the splitting field all I can think to do is say that as is a root of then splits in and so is the splitting field for over .

Any help for this last bit or if you can point out any errors I've already made would be greatly appreciated!

Re: Splitting Fields and Galois Groups

As I posted just thought that to show is also a root of you would say something along the lines of factors into for some and somehow this ends up with as a root... I know this is obviously not correct but am I along the right lines??

Re: Splitting Fields and Galois Groups

one can verify is a root directly:

Re: Splitting Fields and Galois Groups

Haha, cheers; I'm a fool. Other than that does the rest seem OK?