Solution
= Solution
Define $\vartheta(x)=x^p$ for every $x\in K$. In characteristic $p$, all intermediate binomial coefficients vanish, so $\vartheta(x+y)=\vartheta(x)+\vartheta(y)$; multiplicativity and preservation of one are immediate. If $x^p=y^p$, then $(x-y)^p=0$ and the <field> property gives $x=y$. Thus this <field homomorphism> is injective, and finiteness makes it surjective. It is therefore a <field automorphism>, sending $\lambda$ to $\lambda^p$ as required.