= Solution
For nonzero $x\in K$, <Lagrange's theorem for finite groups>[Lagrange's theorem] in $K^\times$ gives $x^{p^n-1}=1$; zero satisfies $x^{p^n}=x$ too. Thus $\vartheta^n$ is the <identity element>[identity], and (iii) excludes every smaller positive order. Hence $\boxed{|\langle\vartheta\rangle|=n}$.
Any <field automorphism> of $K$ fixes $F$ and is determined by its value on $\lambda$. That value must be a root of the <polynomial degree>[degree]-$n$ minimal <polynomial>, so there are at most $n$ automorphisms. The $n$ powers of $\vartheta$ already attain this bound. Consequently $\boxed{\operatorname{Aut}(K)=\langle\vartheta\rangle\cong C_n}$.
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