I have been given a question which was constructing the galois group of $x^3-2$. I am just wondering if my approach is correct, I have found the splitting field, denoted $N$, and found the degree so I know there will be one more automorphism however I am unsure what it would be, the way I worked out the others was looking at the unit circle and seeing which permutations I can take to get back to my original starting point. Please may you read over my work and supply corrections/ help as I have a test coming up. ( Image attached below)
