Galois Group of some polynomials of low degree

Okay, I still having some trouble with finding Galois group.

Let's say that I wish to find the Galois group of over . Now the roots of are where and so the splitting fields of over is . Also the minimal polynomial of is hence we have .

Since the characteristics of we know that . There are only two unique groups (up to isomorphism) of order 4, hence either or .

I feel like it should be . This is my reason (which I know is not right) but I think I am on the right track. Since we have we can write . Now if I would be able to assert that for we must have and .

So in essence I need some like the above that will enable me to eliminate some of the 24 permuatations (unless my argument is totally wrong and it's acutally isomorphic to (Crying) ).

Also, the second problem I was doing is this: find the Galois group of . Since the spltting field of over is (because ). Hence once I solve the above problem I have solved this one as well since they have the same splitting fields, right?

Thanks in advance.