Why does the characteristic of a finite field always have to be prime?

Suppose the characteristic $n$ is composite with factors $1<p,q<n$. Then you could write $$\underbrace{(1+1+\dots+1)}_p\underbrace{(1+1+\dots+1)}_q=\underbrace{1+1+\dots+1}_{n}=0$$ but this means $pq=0$, so $p$ and $q$ are zero divisors, which contradicts the field axioms.

If the characteristic $n = jk \neq 0$ for some $j,k \in \mathbb{N}$ which we are interpreting as the sum of $1$s in the field then $j$ and $k$ are zero divisors and that's important because this means $j^{-1}$ and $k^{-1}$ don't exist, which violates the axiom of a field requiring multiplicative inverses.