Let $K$ and $L$ be extensions of a finite field $F$ of degrees $n$ and $m$,respectively.
How do I show that $KL$ has degree $\mathrm{lcm}(m,n)$ over $F$ and $K\cap F$ has degree $\gcd(m,n)$ over $F$?
Let $K$ and $L$ be extensions of a finite field $F$ of degrees $n$ and $m$,respectively.
How do I show that $KL$ has degree $\mathrm{lcm}(m,n)$ over $F$ and $K\cap F$ has degree $\gcd(m,n)$ over $F$?