Using the tower law with field extensions

Question

Let $L:K$ be a field extension and let $K_1$ and $K_2$ be two intermediate fields such that $L=K(K_1,K_2)$. Show that $[L:K]\leq[K_1:K][K_2:K]^2$.

I know that to show this I will be using the Tower law, with my tower of fields being something like $K\subset K(K_1) \subset K(K_1, K_2) = L$. But I'm not sure how to go about getting $[K_2:K]^2$.

Answer 1

If $L$ is the composite field of $K_1$ and $K_2$, then $[L : K] \leq [K_1 : K][K_2 : K]$. To show this, pick bases $B_1$ and $B_2$ for $K_1$ and $K_2$, then show that $\{\alpha\beta : \alpha \in B_1, \beta \in B_2\}$ is a spanning set for $L$.