Show that the field extension is not Galois.

Question

Let $\alpha$ be complex with $\alpha^2 = \sqrt{3} - \sqrt{5}$. I need to show that $\mathbb{Q}(\alpha)/\mathbb{Q}$ is not a Galois extension.

Any hints would be very welcome.

Answer 1

Try this argument:

The polynomial $f(x)=x^8-16x^4+4$ clearly has precisely four real roots (one for each of the (two) real fourth roots of the zeros of $x^2-16x+4$). Thus the field gotten by adjoining one root of $x^8-16x^4+4$ has four real embeddings, two (pairs of) complex embeddings. But a Galois extension of $\Bbb Q$ has all real embeddings, or all complex.