Normal bases of rings of continuous functions constructed with the (qn)-digit principle

Abstract

When K is a local field with valuation ring V , K. Conrad [6] constructs normal bases of the ring C(V,K) of continuous functions from V to K, using what he calls the extension by q-digit expansion, where q denotes the cardinality of the residue field k of V . In this article, we extend Conrad’s method to the ring C(S,K) of continuous functions from S to K where S denotes a subset of V . Moreover, we no more assume the finiteness of the residue field k, but replace this condition by the precompactness of S. We first recall in Section 1 the notion of normal basis and Conrad’s qdigit principle. In Section 2, we define the extension by (qn)-digit expansion. Then, in Section 3, we generalize Conrad’s q-digit principle by our (qn)-digit principle (Theorem 3.6), that may be applied in particular to Amice’s regular compact subsets [1]. In section 4, we end with several examples.