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?



Examples of perfect pseudo algebraically closed fields in positive characteristic
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.


