In [l] there is given a list of certain open questions in the theory of topological fields. Among them there is A. V. Arkhangel'skii's question: is there a countable nondiscrete topological field without nontrivial convergent sequences? In this paper we give the answer: in any countable field, in any maximal nondiscrete field topology there exist no nontrivial sequences. We mention that from [2] there follows the existence of a countable Abelian topological group without convergent sequences, while in [3] the following theorem is proved: on any Abelian group of cardinality m there exists a precompact topology without convergent sequences.