= Solution
Take $0<m<n$, the intended positive-exponent range. If $\vartheta^m(\lambda)=\lambda$, then $\vartheta^m$ fixes both $F$ and its <field> generator, so it fixes every element of $K=F(\lambda)$. Every element would then be a root of $X^{p^m}-X$. But that <polynomial> has <polynomial degree>[degree] $p^m<p^n=|K|$, impossible for a nonzero <polynomial> over a <field>. Thus $\boxed{\vartheta^m(\lambda)\ne\lambda\quad(0<m<n)}$. The source writes only $m<n$; zero must be excluded, since $\vartheta^0(\lambda)=\lambda$.
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