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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-1-2,
author = {J. E. Marcos},
title = {Locally unbounded topological fields with topological nilpotents},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {21-32},
zbl = {1052.12005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-1-2}
}
J. E. Marcos. Locally unbounded topological fields with topological nilpotents. Fundamenta Mathematicae, Tome 173 (2002) pp. 21-32. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-1-2/