  We study the universal cover of the complex one-dimensional torus as a
model-theoretic structure in a natural language. We consider also abstract
covers of one-dimensional tori over algebraically closed fields of
characteristic zero. The main result states that the structures can be
described uniquely up to isomorphism by a simple (non-first order) sentence,
given a fixed uncountable cardinality of the underlying field. The proof
reduces to the Kummer theory of cyclic Galois extensions over some, not
necessarily finitely generated fields.
