# Infinite groups admitting field structure

Does every infinitely generated Abelian group admit a field structure? (i.e. If $(G,+)$ is an infinitely generated Abelian group, then, is there a binary operation "$\cdot$" such that $(G,+,\cdot)$ is a field?)

Is there any characterization (or classification) of such these groups (admitting field structure) ?

(What about Abelian groups admitting ring structure with nontrivial multiplication?)

-

## migrated from mathoverflow.netMar 2 '14 at 20:59

This question came from our site for professional mathematicians.

## 3 Answers

A field is a vector space over its prime field, so its additive group is either a group of prime exponent $p$, or a torsion-free divisible group, depending on the characteristic. Conversely, any such abelian group is an additive group of an extension of $\mathbb F_p$ or $\mathbb Q$ of appropriate degree.

-
Emil: your answer just provides the explanation of my counterexample. –  Salvatore Siciliano Mar 2 '14 at 18:59

No. Consider for instance the direct product of all groups of distinct prime order.

-

This question is discussed already in MO and you can find the answer in the links below:

-