Your attempts? Have you found the minimal polynomial of $\alpha$? And the conjugates of $\alpha$? Did you realize that $L$ is a biquadratic extension of $\mathbb{Q}$?
– Jack D'AurizioNov 14 '17 at 19:43
The same question was a few hours ago. I gave already an answer with links - see here. Please do not repost the same question twice!
– Dietrich BurdeNov 14 '17 at 19:43
This is not true. You wrote:"Determine $Gal(L/\mathbb{Q})$", i.e., the Galois group of the extension $L/\mathbb{Q}$ (this is not a quotient but rather a field extension).
– Dietrich BurdeNov 14 '17 at 19:48