Inspired by a recent question 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.)

The following paper claims an answer to this question:
You can find it here: http://www.math.uga.edu/~pete/Dicker1966.pdf One might hope for a more aesthetically appealing characterization. I don't know if such a thing has ever been given. 


Another characterization is theorem 2.1 in this paper on the field with one element: http://arxiv.org/pdf/0911.3537 If H is a commutative group, let H+ be H together with a new element 0. To give a field structure on H+ is equivalent to giving a bijection s:H+ > H+ that commutes with all of its conjugatesbyH. Maybe this is similar to Dicker's characterization? Dicker mentions the operation x > 1x, while the s in ConnesConsani is meant to be x > x + 1. 

