Locally unbounded topological fields with topological nilpotents
We construct some locally unbounded topological fields having topologically nilpotent elements; this answers a question of Heine. The underlying fields are subfields of fields of formal power series. In particular, we get a locally unbounded topological field for which the set of topologically nilpotent elements is an open additive subgroup. We also exhibit a complete locally unbounded topological field which is a topological extension of the field of $p$-adic numbers; this topological field is a missing example in the classification of complete first countable fields given by Mutylin.