We study function fields over p-adically closed fields in the first-order language of fields. Using ideas of Duret [D], we show that the field of constants is definable, and that the genus is an elementary property.

We study the first order theory of function fields in the language of fields by using fundamental results on curves in algebraic geometry. We give some applications; for example, using a theorem of G. Cherlin, we prove the undecidability of function fields with nonzero characteristic over an algebraically closed field.