Example of a topological field whose completion is an integral domain but not a field (Q1193250)

Let \(R\) be a countable, Noetherian unique factorization domain (for example \(\mathbb{Z})\) with a prime element \(p\). Consider the domain \(A:=R[X]_{(p,X)}\). The nonzero ideals of \(A\) form a neighborhood base at zero for a field topology on the quotient field \(L\) of \(A\). It is proved that the completion of \(L\) is an integral domain that is not a field. This answers a question posed by \textit{W. Wiȩsław} [Topological fields (1988; Zbl 0661.12011) p. 251, Problem 3].