Define a $K$-isomorphism, $\varphi: L \rightarrow L^{\prime}$, where $L, L^{\prime}$ are fields containing a field $K$, and define $\operatorname{Aut}_{K}(L)$.

Suppose $\alpha$ and $\beta$ are algebraic over $K$. Show that $K(\alpha)$ and $K(\beta)$ are $K$-isomorphic via an isomorphism mapping $\alpha$ to $\beta$ if and only if $\alpha$ and $\beta$ have the same minimal polynomial.

Show that $\operatorname{Aut}_{K} K(\alpha)$ is finite, and a subgroup of the symmetric group $S_{d}$, where $d$ is the degree of $\alpha$.

Give an example of a field $K$ of characteristic $p>0$ and $\alpha$ and $\beta$ of the same degree, such that $K(\alpha)$ is not isomorphic to $K(\beta)$. Does such an example exist if $K$ is finite? Justify your answer.