I have the following question:
Let E= Q(w, $\sqrt2$), where w= $e^\frac{2i\pi}{3}$
I need to show that E:Q is a Galois Extension, compute [E:Q], find the elements of the Galois group, and determine the isomorphism.
However, I am having a lot of trouble figuring out how to simplify w into a square root. I am seeing some people say that w is equal to $\sqrt-3$, whereas others say w is equal to $\sqrt[3]{2}$. Once I can simplify E, I think I will be able to get started, but any help would be greatly appreciated.
As $\omega=\frac12(-1+i\sqrt3)$ then $E=\Bbb Q(\sqrt2,i\sqrt3)$. This is the splitting field of $(X^2-2)(X^2+3)$ over $\Bbb Q$. As a splitting field and since $\Bbb Q$ has characteristic zero, $E/\Bbb Q$ is a Galois extension.