Separably closed fields

SCF is the theory of separably closed fields. So a field $$K$$ is a model of SCF if every polynomial $$P(X) \in K[X]$$ whose roots in $$K^{alg}$$ are distinct, has at least one root in $$K$$. Equivalent conditions are that every irreducible polynomial over $$K$$ is of the form $$X^{p^k} - a$$ where $$p$$ is the characteristic, or that the perfect closure of $$K$$ is algebraically closed. In characteristic zero, SCF is the same as ACF. So one usually restricts to the case of positive characteristic.

If $$K$$ is a separably closed field of characteristic $$p$$, then $$K^p$$ (the set of $$p$$th powers) is a subfield of $$K$$. The degree of the extension $$K/K^p$$ is $$p^n$$, where $$n$$ is a number called the degree of imperfection. The degree of imperfection of the separable closure of $$K(t_1,\ldots,t_n)$$ is $$n$$, if $$K$$ is an algebraically closed field of characteristic $$p$$. It turns out that models of SCF are determined up to elementary equivalence by their characteristic and degree of imperfection (as an element of $$\{0,1,2,\ldots,\infty\}$$).

All models of SCF are stable, and the non-algebraically closed ones are strictly stable, i.e., not superstable.

SCF does not have quantifier elimination in the language of rings. However, it does have quantifier elimination or model completeness in a language where one names a "p-basis" and adds "lambda functions." Basically, it turns out that if $$K$$ has degree of imperfection $$n$$, then one can find elements $$a_1, \ldots, a_n \in K$$ such that the set of monomials of the form $$\prod_{i = 1}^n a_i^{j_i} \quad 0 \le j_i < p$$ is a vector space basis of $$K$$ over $$K^p$$. Then each element $$x \in K$$ has a unique expression of the form $$x = \sum_I (\lambda_I(x))^p a^I$$ where $$I$$ ranges over multiindices in $$\{0,\ldots,p-1\}^n$$. After naming the elements of a $$p$$-basis, the $$\lambda_I$$'s become definable functions from $$K$$ to $$K$$, and it turns out that after adding symbols for them, one has quantifier elimination, or maybe just model completeness.

When the degree of imperfection is infinite, something more complicated must be done.

Separably closed fields played a key role in Hrushovski's proof of Geometric Mordell Lang in positive characteristic.