We consider separably closed fields of characteristic $p > 0$ and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the $p$-component functions.
"The theory of modules of separably closed fields. I." J. Symbolic Logic 67 (3) 997 - 1015, September 2002. https://doi.org/10.2178/jsl/1190150144