The frame of the p-adic numbers

Abstract

The algebraic nature of a frame allows its definition by generators and relations. Joyal used this to introduce the frame of the real numbers and Banaschewski studied this frame with a particular emphasis on the pointfree extension of the ring of continuous real functions; he provided a pointfree version of the Stone-Weierstrass theorem. In this paper, we explore this situation for the case of the p-adic numbers; we define the frame of the p-adic numbers by generators and relations. The field of the p-adic numbers Qp is the completion of Q with respect to the p-adic absolute value |⋅|p, which satisfies |x+y|p≤max⁡{|x|p,|y|p}. Dieudonné proved that the ring Qp[X] of polynomials with coefficients in Qp is dense in the ring C(F,Qp) of continuous functions defined on a compact subset F of Qp with values in Qp, and Kaplansky extended this result by proving that if F is a nonarchimedean valued field and X is a compact Hausdorff space, then any unitary subalgebra A of C(X,F) which separates points is uniformly dense in C(X,F). We give a p-adic version of this theorem in pointfree topology.