Is $\operatorname{Gal}(\overline{\mathbb{F}_p} \,/\, \mathbb{F}_p)$ cyclic?

Question

I know if $L/K$ is a extension of finite fields then $\text{Gal}(L/K)=\langle \phi \rangle$ where $\phi:L\longrightarrow L$ is the Frobenius automorphism.

How can I show that it is also true for the extension $\overline{\mathbb{F}_p}\,/ \,\mathbb{F}_p$? Because in this case $\overline{\mathbb{F}_p}$ is not a finite field.