I had this homework problem, I think I got it correct, just want a check from somebody.
Find the Galois group of over .
I get that the splitting field is where .
The next step is to calculate the dimension of this field. Here I used: let be irreducible over with degree and , then is irreducible over if . Since is minimal polynomial for it follows from above that this polynomial is irreducible over . Thus, . And thus, .
Now let , .
Thus, and , also .
Finally, is a set of distinct -automorphism.
There are all together. Thus we have found all elements of .
My remaining question is what type of group is this?
I think this is the Frobenius group.
And what is the subgroup diamgram of this group?
(Wikipedia did not have it).