Basis for $\mathbb Q (\sqrt2 , \sqrt3 )$ over $\mathbb Q$

One can use the fact that $\mathbb{Q}(\sqrt2,\sqrt3)$=$(\mathbb{Q}(\sqrt2))(\sqrt3)$ the latter of which has elements of the form $a+b\sqrt3$ where $a,b\in \mathbb{Q}(\sqrt2)$ since $[(\mathbb{Q}(\sqrt2))(\sqrt3) : \mathbb Q(\sqrt2)]=2$. From here we observe that $a=a_1+a_2\sqrt2$ and $b=b_1+b_2\sqrt2$ for some rationals $a_i$ and $b_i$ since $[\mathbb Q(\sqrt2) : \mathbb Q]=2$ . Together substituting back into $a+b\sqrt3$ you can see that $\{1, \sqrt2 , \sqrt3, \sqrt6 \}$ is a spanning set. Since the dimension of $\mathbb{Q}(\sqrt2,\sqrt3)$ as a vector space over $\mathbb{Q}$ is $4$ we know that the spanning set is also a basis as desired.

You can note that $\sqrt{3}$ has degree $2$ over $\mathbb{Q}(\sqrt{2})$, because $$ (a+b\sqrt{2})^2=3 $$ leads to $$ a^2+2b^2=3,\qquad 2ab=0 $$ which has no solution in the rational numbers. Therefore $\{1,\sqrt{3}\}$ is a basis of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ over $\mathbb{Q}(\sqrt{2})$, which in turn has $\{1,\sqrt{2}\}$ as a basis over $\mathbb{Q}$. The standard proof of the dimension theorem says that $$ \{1\cdot 1,1\cdot\sqrt{2},1\cdot\sqrt{3},\sqrt{2}\cdot\sqrt{3}\} $$ is a basis of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ over $\mathbb{Q}$.