Thread: Galois Group

I'm trying to find the Galois Group of the splitting field of over Q.

This polynomial is irreducible, and the splitting field is generated by =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 over Q, and that can't be correct.

Originally Posted by robeuler

I'm trying to find the Galois Group of the splitting field of over Q.

This polynomial is irreducible, and the splitting field is generated by =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 over Q, and that can't be correct.

You can go here, it may be helpful.

Originally Posted by ThePerfectHacker

You can go here, it may be helpful.

so ?

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.