List a basis for $\mathbb K =\mathbb Q (\sqrt2 , \sqrt3 )$ as a vector space over $\mathbb Q $.
I don't know how people come to the conclusion of a claimed basis. Like I am pretty sure that we just claim $\{1, \sqrt2 , \sqrt3, \sqrt6 \}$ is a basis and then prove that it is LI and it is a spanning set of $\mathbb K$ but how do you even think of this claim? That is my first question. I know that the dimension of $\mathbb K$ is $4$ so the basis should have $4$ elements but it doesn't really tell you which $4$ specifically.
Secondly, to prove that these are LI:
$1$ is LI to all the other elements clearly since the others are irrational and $1$ is rational. Not sure on how the other three are LI to each other though.
Also I am unsure on the spanning part.
Best Answer
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.