The field of rational numbers is algebraically closed in some of its topological completions (Q1295550)

The author constructs a class of examples of locally unbounded ring topologies on \(\mathbb{Z}\) (resp. \(k[X]\), \(k\) a field). Similar constructions of field topologies on \(\mathbb{Q}\) and \(k(X)\) are presented. In the case of rings, the completion is an integral domain and it is a field in case of field topologies. It is not clear whether the topologies are minimal or not. Remark: The author's construction can be directly extended to finite field extensions \(K\) of \(\mathbb{Q}\), for which the integral closure of \(\mathbb{Z}\) in \(K\) is UFD.