Solution
= Solution
The <multiplicative group of a field>[multiplicative group] $K^\times$ has order $p^n-1$, so <Lagrange's theorem for finite groups> gives $x^{p^n-1}=1$ for $x\ne0$. Including zero, $x^{p^n}=x$ for every $x\in K$. Therefore $\vartheta^n$ is the identity. No positive smaller power is the identity by part (iii), so
$$
\boxed{|\vartheta|=n}.
$$
For $n=1$ this says that Frobenius itself is the identity, as it should be.