= Solution
Define $\vartheta(x)=x^p$ on all of $K$. The binomial coefficients between the endpoints are divisible by $p$, so $(x+y)^p=x^p+y^p$; also $(xy)^p=x^py^p$ and $1^p=1$. Thus this is a <field> homomorphism. Its kernel is zero, and an <injective> self-map of the finite set $K$ is <surjective>. Hence $\boxed{\vartheta\text{ is an automorphism}}$, the <Frobenius automorphism>, with the required image of $\lambda$. It fixes $F$ by the preceding <prime>-subfield argument.
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