  We show that algebraic analogues of universal group covers, surjective group
homomorphisms from a $\mathbb{Q}$-vector space to $F^{\times}$ with "standard
kernel", are determined up to isomorphism of the algebraic structure by the
characteristic and transcendence degree of $F$ and, in positive characteristic,
the restriction of the cover to finite fields. This extends the main result of
"Covers of the Multiplicative Group of an Algebraically Closed Field of
Characteristic Zero" (B. Zilber, JLMS 2007), and our proof fills a hole in the
proof given there.
