Given fields: $A$ = $\Bbb{Q}$, $B = \Bbb{Q}(x)$, and $C = \Bbb{Q}(x,y)$, with $x$ = $(-1)^{2/3}$ and $y$ = $2^{1/3}$, what would be the field bases for $C:A$ and $B:A$ ? I got the basis for $B:A$ to be degree 2 as the minimal polynomial for $x$ is $x^{2}+x+1=0$, so the basis for would ${1,x}$. As for $C:A$, I decided that $C:B$ would be degree 3, since the minimal polynomial would be $x^{3}-2=0$, so by tower theorem $C:A$ degree would be 6, giving me a basis of $1,x,y,xy,y^{2}x,y^{2}$, with the hopes that these are all linearly independent in $\Bbb{Q}$. Is this reasoning/any of this correct?
Assuming this is correct, would the galois groups of $C:A$ and $B:A$ just be the permutations of (+/-) of the bases?