# 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) ?

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

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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.