  Siegel proved that every totally positive element of a number field K is the
sum of four squares, so in particular the Pythagoras number is uniformly
bounded across number fields. The p-adic Kochen operator provides a p-adic
analogue of squaring, and a certain localisation of the ring generated by this
operator consists of precisely the totally p-integral elements of K. We use
this to formulate and prove a p-adic analogue of Siegel's theorem, by
introducing the p-Pythagoras number of a general field, and showing that this
number is uniformly bounded across number fields. We also generally study
fields with finite p-Pythagoras number and show that the growth of the
p-Pythagoras number in finite extensions is bounded.
