degree of finite extension of finite field

Question

Given a field extension $\mathbb{F}_{p^m}$ over $\mathbb{F}_p$, I'm trying to prove that $[\mathbb{F}_{p^m}:\mathbb{F}_p]=m$.
Important note: I'm trying to prove it without using the fact that $Gal(\mathbb{F}_{p^m}/\mathbb{F}_p)$ is cyclic group of order $m$.