Prove the following:

**(i)** If $\displaystyle \{a_1,\ldots,a_r\}\subset L$ is linearly independent over $\displaystyle K$ , which implies $\displaystyle r\leq (L:K)$

and

$\displaystyle \{b_1,\ldots,b_s\}\subset K$ is linearly independent over $\displaystyle F$ , which implies $\displaystyle s\leq (K:F)$

then,

the $\displaystyle rs$ elements $\displaystyle a_ib_j$ of $\displaystyle L$ are linearly independent over $\displaystyle F$ .

**(ii)** Those $\displaystyle rs$ elements of $\displaystyle L$ generate $\displaystyle L$ as a vector space over $\displaystyle F$ .

This will lead to:

$\displaystyle (L:F)=(L:K)(K:F)$

Fernando Revilla
Edited: Sorry, I read too quickly and my hints were for a different problem, although related with it.