base of the vectorspace $\Bbb{Q}(a)$ over $\Bbb{Q}$ i have to find the base of the vectorspace $\Bbb{Q}(\alpha)$ over $\Bbb{Q}$ with $\alpha = \sqrt{1+\sqrt{3}}$.

i have found the minimal polynom of $\alpha$ over $\Bbb{Q}$: $f(x)=x^{4}-2x^{2}-2$.
The degree of this polynom is 4 so dim$\Bbb{Q}(\alpha) = 4  $.
Is this correct?

can i take as base 1,$\alpha$,$\alpha^{2}$,$\alpha^{3}$?
Yes you can although there other forms of bases.
Yes you can although there other forms of bases.