The Weierstrass theorem in fields with valuations

Abstract

In [2, Theorem 32]2 the author showed that an analogue of the Weierstrass-Stone theorem holds in topological rings having ideal neighborhoods of 0. Earlier, Dieudonne [1] had proved the Weierstrass-Stone theorem for the field of p-adic numbers. Now the field of p-adic numbers has an open subring (the p-adic integers) with ideal neighborhoods of 0. It seems plausible, therefore, to expect that the method of [2] will apply, provided one has a supplementary device for getting into the p-adic integers. This is in fact the case, and the result is applicable to any division ring with a valuation of rank one.3 The requisite lemma reads as follows: