Which theorem tells about smallest field containing two given fields?

Question

Suppose $\mathbb{F} _{p^n}$ and $\mathbb{F} _{p^m}$ are two finite fields where p is a prime number and n,m$\in \mathbb{N}$, what is the smallest field containing these fields ?

Answer 1

$\Bbb F_{p^k}\subseteq\Bbb F_{p^h}$ if and only if $k\mid h$. Therefore you are looking for $\Bbb F_{p^{\operatorname{lcm}(n,m)}}$.