1. ## Galois Group

I'm trying to find the Galois Group of the splitting field of $x^{8}-3$ over Q.

This polynomial is irreducible, and the splitting field is generated by $\sqrt[8]{3}$=a and w, any primitive 8th root of unity.

Here I am a bit uncertain. I believe like Q(a) is an extension of degree 8. And Q(w)=Q(i) is an extension of degree 2. So would Q(a,w) be an extension of degree 16. But this is the exact same as $x^{8}-2$ over Q, and that can't be correct.

2. Originally Posted by robeuler
I'm trying to find the Galois Group of the splitting field of $x^{8}-3$ over Q.

This polynomial is irreducible, and the splitting field is generated by $\sqrt[8]{3}$=a and w, any primitive 8th root of unity.

Here I am a bit uncertain. I believe like Q(a) is an extension of degree 8. And Q(w)=Q(i) is an extension of degree 2. So would Q(a,w) be an extension of degree 16. But this is the exact same as $x^{8}-2$ over Q, and that can't be correct.
You can go here, it may be helpful.

3. Originally Posted by ThePerfectHacker
You can go here, it may be helpful.
so $[Q(a,w):Q]=\varphi(3)*16=32$?

I am very confused about how to calculate the Galois group if this is true. I know it is of order 32, but i don't understand how the roots can be permuted.