Past exam of the Probability, statistics and options II course of the University of Oxford / 2009 / A / A2 / iii

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 degree $p^m<p^n=|K|$, impossible for a nonzero polynomial over a field. Thus $\boxed{{\vartheta }^m(\lambda )\ne \lambda (0<m<n)}$. The source writes only $m<n$; zero must be excluded, since ${\vartheta }^0(\lambda )=\lambda$.