Inspired by [a recent question](https://mathoverflow.net/questions/12873/split-powers-of-the-multiplicative-group-of-a-field) on the multiplicative group of fields. Necessary conditions include that there are at most $n$ solutions to $x^n = 1$ in such a group and that any finite subgroup is cyclic. Is this sufficient? (Edit: Well, no, it's not, since the only such groups which are finite are the cyclic groups of order one less than a prime power. Hmm.)



Which commutative groups are the group of units of some field?
The following paper claims an answer to this question:



> 
> Dicker, R. M.
> A set of independent axioms for a field and a condition for a group to be the multiplicative group of a field.
> Proc. London Math. Soc. (3) 18 1968 114--124.
> 
> 
> 


You can find it here:


[http://alpha.math.uga.edu/~pete/Dicker1966.pdf](http://alpha.math.uga.edu/%7Epete/Dicker1966.pdf)


One might hope for a more aesthetically appealing characterization. I don't know if such a thing has ever been given.


