= Solution
Here the iteration index is a positive <integer>: $1\leq m<n$. If $\vartheta^m(\lambda)=\lambda$, then $\vartheta^m$ fixes $F$ and the generator, so it fixes all of $K$. Every element of $K$ would then solve $T^{p^m}-T=0$. A nonzero <polynomial> of degree $p^m$ over a <field> has at most $p^m$ roots, contradicting $|K|=p^n>p^m$. Thus \b[$\vartheta^m(\lambda)\ne\lambda$]. The printed inequality alone omits positivity of $m$; at $m=0$ the claimed inequality would be false.
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