Is the extension $ℚ(\alpha):ℚ(\alpha^2)$ Galois? (for $\alpha$ a complex number)
I understand to show an extension is Galois one must show it is normal and separable.
I've never seen an example like this before so I don't know where to start with the proof. Does anyone have any links to similar questions or sites which may help me understand and solve this problem?
Thank you.
A finite field extension $L/K$ is Galois if and only if $L$ is a splitting field of some separable polynomial $f\in K[x]$. In your example, this is the case because $\mathbb{Q}(\alpha)$ is the splitting field of $x^2-\alpha^2\in\mathbb{Q}(\alpha^2)[x]$. Well, at least this polynomial is separable if $\alpha\ne 0$. If $\alpha=0$ then it is a trivial extension, so it is obviously Galois.