SCF is the theory of separably closed fields. So a field is a model of SCF if every polynomial whose roots in are distinct, has at least one root in . Equivalent conditions are that every irreducible polynomial over is of the form where is the characteristic, or that the perfect closure of is algebraically closed. In characteristic zero, SCF is the same as ACF. So one usually restricts to the case of positive characteristic.
If is a separably closed field of characteristic , then (the set of th powers) is a subfield of . The degree of the extension is , where is a number called the degree of imperfection. The degree of imperfection of the separable closure of is , if is an algebraically closed field of characteristic . It turns out that models of SCF are determined up to elementary equivalence by their characteristic and degree of imperfection (as an element of ).
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 has degree of imperfection , then one can find elements such that the set of monomials of the form is a vector space basis of over . Then each element has a unique expression of the form where ranges over multiindices in . After naming the elements of a -basis, the 's become definable functions from to , 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.