Galois group Abstract algebra

$omega$ is the primitive $4^{th}$ root of unity, then find $Galleft(mathbb{Qleft(sqrt{2},omegaright)/mathbb{Q}}right).$

First i calculate order of $Galleft(mathbb{Qleft(sqrt{2},omegaright)/mathbb{Q}}right)$ which is $4$ by use of minimal polynomial.

$left[mathbb{Q}left(sqrt{2},omegaright):mathbb{Q}right]=left[mathbb{Q}left(sqrt{2},omegaright):mathbb{Q}left(omegaright)right]left[mathbb{Q}left(omegaright):mathbb{Q}right]$

$left[mathbb{Q}left(sqrt{2},omegaright):mathbb{Q}left(omegaright)right]=2$

$left[mathbb{Q}left(omegaright):mathbb{Q}right]=2$ because $x^2+1$ is minimal polynomial.

Therefore $left[mathbb{Q}left(sqrt{2},omegaright):mathbb{Q}right]=2times 2=4$.

(I am not sure that i calculate right)

I don’t know how elements of $Galleft(mathbb{Q}left(sqrt{2},omegaright)/mathbb{Q}right)$ look like.

I think it is of the type

$varepsilon=omega rightarrow omega,~sqrt{2}rightarrow sqrt{2}$

$alpha=omegarightarrow omega^2,~sqrt{2}rightarrow sqrt{2}$

$beta=omegarightarrow omega^3,~sqrt{2}rightarrow sqrt{2}$

$gamma=omega rightarrow omega,~sqrt{2}rightarrow omegasqrt{2}$.

Four elements complete here and i think i did wrong.

Any hint how to find element.

Thank you.