Every positive rational number is the sum of four squares by a well-known theorem of Euler. As predicted by Hilbert and proven by Siegel, this generalizes to arbitrary number fields K when one replaces 'positive' by 'totally positive', i.e. positive with respect to every embedding of K into the reals. I will motivate and present a p-adic analogue of this, which gives a constructive description of those elements of K that are totally p-adically integral, i.e. p-adic integers for each embedding of K into the p-adic numbers. The proof of this result involves the Brauer-Hasse-Noether local-global principle for central simple algebras. Joint work with Sylvy Anscombe and Philip Dittmann.