Find the Galois group of .
I could have this wrong, but here goes:
The splitting field for is , where is a third root of unity.
We then have
Since we can determine all automorphisms based on the roots of and , where is another root of , let's see where that takes us.
Define the automorphisms as follows:
Now note , so .
I'll leave it to you to see why .
So I'm thinking , where is a three-cycle.