How to Determine Galois Groups and Subfields?

[minibear]

I have trouble in determining Galois group.
Can anyone help me with the following question:

how to determine the Galois group of (x^2-2)(x^2-3)(x^2-5), determine all the subfields of the splitting field of this polynomial?

how to determine the elements of the galois group of x^p-2 for p is prime.

Thanks a lot!

 

[HallsofIvy]

The set of solutions of x^p= a are of the form $|a|^{1/p}\omega^i$ with i ranging from 0 to p-1, where $\omega$ is the "principal pth root of unity". The Galois group is the permutation group of that set: the permutation group on p objects.