The stable-field and simple-field conjectures

Recall that a field $K$ is stable if its first-order theory is stable, and simple if its first-order theory is simple. A field is separably closed if it has no proper finite separable algebraic extension, and it is bounded if it has only finitely many extensions of each finite degree. A field is pseudoalgebraically closed ($\mathrm{PAC}$) if every geometrically integral $K$-variety has a $K$-point.

Stable-field and simple-field conjectures.

\begin{enumerate} \item \text{Infinite stable fields are separably closed.} \item \text{Infinite simple fields are bounded }\mathrm{PAC}. \end{enumerate}

These are presented as longstanding open conjectures about the relationship between model-theoretic tameness and the arithmetic of fields. Separably closed fields are stable, while bounded $\mathrm{PAC}$ fields are simple; the conjectures assert converses in the infinite case.