The specific question I'm having trouble with is:
Show X^5-4X+2 has Galois Group S_5 over Q. (I've done this). What is the Galois Group of the polynomial over Q(i)?
I can't do the second part. I'm pretty sure the polynomial is irreducible over Q(i), but that's as much as I can do.
Any help or general tips for dealing with these situations would be appreciated.