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What would be the minimal extension of $F_{p}$ that would have exactly one automorphism?

I am looking for the minimum extension of $\mathbb F_{p}$ such that $Aut(\mathbb F_{p})\subseteq Aut(\mathbb E/\mathbb F_{p})$

By elementary Galois theory, every subfield of an algebraic extension field over $\mathbb F_{p}$ has a unique automorphism. However I was wondering if a minimal extension such as $\mathbb E/\mathbb F_{p}$ would do so.

The only algebraic extension of a field $F$ is $F$.

Let $\phi \in \text{Aut}(F)$. Since $F$ is algebraic, there exists $\varphi \in \text{Gal}(F/\mathbb{F}_p)$ such that $\phi = \varphi \circ \varphi^{ - 0b46394aab

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