Continuous functions on discrete valuation rings
Let \(R\) be a complete discrete valuation ring with field of fractions \(K\) and maximal ideal \({\mathfrak p}\). The author gives an explicit and simple construction of an \(R\)-base \(\{\varphi_n\), \(n=0,1,2\dots\}\) of the \(R\)-module generated by the set of all polynomial functions from \(R\) to \(R\). Using this base, he shows that when \(R/{\mathfrak p}\) is a finite field, then a function \(f\) from \(R\) into any finite extension \(F\) of \(K\) is continuous if and only if there exist (uniquely determined) elements \(a_n\in F\) such that \(\lim_{n\to\infty} a_n=0\) and \(f(a)= \sum_{n=0}^\infty a_n \varphi_n(a)\) for all \(a\) in \(R\). The above result is a generalization of Mahler's expansion theorem.
scientific article; zbMATH DE number 1274809
scientific article