Is there any known example of a perfect pseudo algebraically closed field of positive characteristic containing $\overline{\mathbb{F}_p}$ but is not algebraically closed?
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$\begingroup$ You can "formally" produce such examples. Begin with any finitely generated extension $K/k$ of an algebraically closed field $k$ of characteristic $p$. Consider the filtered system of function fields $L/K$ of geometrically irreducible $K$-varieties, partially ordered by existence of embeddings $L / K < L' / K$ if there exists a $K$-embedding of $L$ in $L'$. The colimit of all of these field extensions is a field extension $E/K$ such that for every $L/E$ as above, $L$ is purely inseparable over $E$. Now replace $E$ by its perfect closure. Note: $K$ is already separably closed in $E$. $\endgroup$– Jason StarrCommented Nov 13, 2015 at 0:51
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$\begingroup$ In that construction, I had the partial order wrong: $L/K < L'/K$ if there exists a $K$-embedding of $L$ in $L'$ such that $L$ is separably closed in $L'$. $\endgroup$– Jason StarrCommented Nov 13, 2015 at 3:58
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$\begingroup$ By the way, Fried-Jarden proved that every perfect pseudo algebraically closed field that contains $\overline{\mathbb{F}}_p$ (or equivalently, all roots of unity) is quasi-algebraically closed: all specializations of Fano hypersurfaces have rational points. I extended this: every specialization over such a field of a separably rationally connected variety has a rational point. Ax's conjecture says that this should be true even if the field does not contain $\overline{\mathbb{F}}_p$. $\endgroup$– Jason StarrCommented Nov 14, 2015 at 12:42
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Yes, every ultraproduct $F$ of fields of the form $\mathbb{F}_{p^{n!}}$ has this property. It is pseudo finite and hence perfect and pseudo algebraically closed. Moreover, since every polynomial with coefficients in $\mathbb{Z}$ splits in $\mathbb{F}_{p^{n!}}$ for sufficiently large $n$, the field $F$ contains an algebraic closure of the prime field by Los' theorem.
