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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?)

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3 Answers 3

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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.

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  • $\begingroup$ Emil: your answer just provides the explanation of my counterexample. $\endgroup$
    – Salvatore Siciliano
    Commented Mar 2, 2014 at 18:59
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No. Consider for instance the direct product of all groups of distinct prime order.

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