# Thread: A simple problem in algebra from feilds

1. ## A simple problem in algebra from feilds

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] .

If $F\subseteq K\subseteq L$ and $K/L$ is finite then $L/K$ and $K/F$ is finite. Furthermore, $[L:F] = [L:K][K:F]$. Therefore, $[K:L]$ divides $[L:F]$. To see this, note if $\{ a_1,...,a_n\}$ is basis for $K/F$ and $\{b_1,...,b_m\}$ is basis for $L/K$ then $\{ a_ib_j\}$ is a basis for $L/F$.