ra.rings and algebras - Which commutative groups are the group of units of some field?

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 conjugates-by-H.

Maybe this is similar to Dicker's characterization? Dicker mentions the operation x --> 1-x, while the s in Connes-Consani is meant to be x --> x + 1.