Geometric reasons finite fields have prime power orders? All variations of proofs that finite fields have prime power orders have a very algebraic feel to them. I was wondering - is there a more geometric way to see why this is true?

A finite field is a vector space over some $\mathbb{F}_p$, so it has a basis and its elements are in bijection with some $n$-uples of coordinates in $\mathbb{F}_p$, which tells you it has cardinal $p^n$. I don't know if you consider that a geometric or algebraic argument though.
