On the Riemann \(\mathfrak p\)-space of a field. The \(\mathfrak p\)-adic analogue of Weierstrass' approximation theorem and related problems
Let \(F\vert K\) be a field extension. Suppose that the base field \(K\) is equipped with a place \(\mathfrak p\) (trivial or nontrivial), with residue field \(k\). The Riemann \(\mathfrak p\)-space of \(F\vert K\) consists of those extensions \(\mathfrak P\) of \(\mathfrak p\) to \(F\) which have the same residue field, namely \(k\). Every element \(f\in F\) defines naturally a function \(f\colon S_{\mathfrak p}\to k\cup \infty\), assigning to \(\mathfrak P \in S_{\mathfrak p}\) the image \(f(\mathfrak P)\in k\cup \infty\). Consider the ring \(I_{\mathfrak p}\) of those \(f\in F\) whose corresponding functions are holomorphic, i.e. \(f(\mathfrak P)\ne \infty\) for all \(\mathfrak P \in S_{\mathfrak p}\). Now if the field \(k\) carries a topology then this induces naturally a topology on the Riemann \(\mathfrak p\)-space \(S_{\mathfrak p}\), and the holomorphic functions from \(L_ {\mathfrak p}\) are continuous. Question: Is Weierstrass approximation theorem true for the ring of holomorphic functions? In other words: Is every continuous function \(\varphi\colon: S_{\mathfrak p} \to k\) (uniformly) approximable by holomorphic functions \(f \in L_{\mathfrak p}\)? In the present paper it is shown that the answer is affirmative, provided the residue field \(k\) is of one of the following types: (I) \(k\) is finite. (II) \(k\) is the field of formal power series over a finite field. (III) \(k\) is the field \(\mathbb Q_p\) of rational \(p\)-adic numbers or a finite extension of that field (for some prime number \(p\)). Applications are given to the theory of formally \(p\)-adic fields. In particular, the explicit construction of totally \(p\)-integral elements is discussed in detail. Remark by the author: Since each of the conditions (I) -- (III) imply that \(S_{\mathfrak p}\) is compact, the validity of our Weierstrass approximation theorem can also be deduced from the paper of \textit{D. G. Cantor} [Pac. J. Math. 21, 473--478 (1967; Zbl 0146.37301)]. See also the more general theorem: \textit{P. R. Chernoff}, \textit{R. A. Rasala} and \textit{W. C. Waterhouse} [ibid. 27, 233--240 (1968; Zbl 0172.41402)].
scientific article; zbMATH DE number 3595285