The nilpotency of the nil metric 𝔽-algebras

Let [Formula: see text] be a normed field. In this work, we prove that every nil complete metric [Formula: see text]-algebra is nilpotent when [Formula: see text] has characteristic zero. This result generalizes Grabiner’s Theorem for Banach algebras, first proved in 1969. Furthermore, we show that a metric [Formula: see text]-algebra [Formula: see text] and its completion [Formula: see text] satisfy the same polynomial identities, and consequently, if [Formula: see text] and [Formula: see text] is nil, then [Formula: see text] is nilpotent. Our results allow us to resolve Köthe’s Problem affirmatively for complete metric algebras over normed fields of characteristic zero.