The real/p-adic closures of an ordered/p-valued field need not be complete. Conversely, one may wonder when an ordered/p-valued field is dense in its real/p-adic closures.
We study the property of a field that it is dense in all its real/p-adic closures. We examine when this property is elementary in the language of rings. It is not always elementary, but for fields with finite Pythagoras/p Pythagoras numbers, it is so. In particular we show that this property holds for models of the theory of algebraic fields of characteristic zero. This essentially includes all previously known examples. This is joint work with Philip Dittmann and Arno Fehm. | 
