Index notation meaning

Question

So if we have something like $[\mathbb Q (\sqrt2 , \sqrt3 ) : \mathbb Q ]=a$, is the value of this the actual degree of $\mathbb Q (\sqrt2 , \sqrt3 ) $? Or is it the least value of the degree? So does it mean the degree is at least $a$?

Answer 1

$[\mathbb{Q}(\sqrt{2},\sqrt{3}): \mathbb{Q}]$ is the dimension of $\mathbb{Q}(\sqrt{2},\sqrt{3})$ as a $\mathbb{Q}$-vector space, which in this case is equal to $4$.
As for "at least", if $F,L,K$ are fields such that $F \subseteq L \subseteq K$, then we have $$[K:F]=[K:L][L:F],$$ so that we obviously have $[K:F] \geq [L:F]$ as all terms are $\geq 1$. Which means that if you have some extension $K$ of $F$ which has a subfield $L$ which is also an extension $F$ you can say that degree of $K$ over $F$ is at least the degree of $L$ over $F$. And in particular for any $\alpha \in K$ we have $[K:F]=[K:F(\alpha)][F(\alpha):F]$.