Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic

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.