I need a rigourous proof for the following problem if possible:-
Let L be a finite extension of F and let K be a subfeild of L such that it contains F.
Show that [K:F] | [L:F] .
Please help.
If $\displaystyle F\subseteq K\subseteq L$ and $\displaystyle K/L$ is finite then $\displaystyle L/K$ and $\displaystyle K/F$ is finite. Furthermore, $\displaystyle [L:F] = [L:K][K:F]$. Therefore, $\displaystyle [K:L]$ divides $\displaystyle [L:F]$. To see this, note if $\displaystyle \{ a_1,...,a_n\}$ is basis for $\displaystyle K/F$ and $\displaystyle \{b_1,...,b_m\}$ is basis for $\displaystyle L/K$ then $\displaystyle \{ a_ib_j\}$ is a basis for $\displaystyle L/F$.