= Solution
Take $p=11$. Write $Y=X^5$, so the extension is
$$
\mathbb F_{11}(X)/\mathbb F_{11}(Y).
$$
The polynomial $t^5-Y$ is irreducible over $\mathbb F_{11}(Y)$, for example by Eisenstein's criterion at $Y$ in $\mathbb F_{11}[Y]$, and has $X$ as a root. Thus the extension has degree five.
The element $3\in\mathbb F_{11}^{\times}$ has order five. Hence all roots of $t^5-Y$ are
$$
X,\ 3X,\ 3^2X,\ 3^3X,\ 3^4X,
$$
and already lie in $\mathbb F_{11}(X)$. The extension is therefore the splitting field of $t^5-Y$ and is normal. Its derivative is $5t^4\ne0$ in characteristic eleven, so the polynomial is separable. By part (c),
$$
\boxed{\mathbb F_{11}(X)/\mathbb F_{11}(X^5)\text{ is Galois}.}
$$
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