TITLE: Which commutative groups are the group of units of some field?
QUESTION [16 upvotes]: 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.)

REPLY [7 votes]: 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.