# Group algebra ever a field?

Let GGG be a group. Let KKK be any field and consider its group algebra K[G]K[G]. It is known that if GG is a finite group then K[G]K[G] is never a field. Or even if GG is not finite: if GG has at least one non-trivial torsion element, then K[G]K[G] is not a field.

Question: Do there exist infinite groups GG such that K[G]K[G] is a field or a division ring?

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When |G|>1|G|>1 the augmentation ideal is nontrivial, so it is never a simple ring, much less a field or division ring.

The augmentation ideal is the kernel of the ring map ∑g∈Grgg↦∑g∈Grg\sum_{g\in G} r_gg\mapsto \sum_{g\in G}r_g. It is nontrivial, for example, because 1−g1-g is in it for any nonidentity element g∈Gg\in G.

So, if KK is a field, the only way for K[G]K[G] to be a division ring is if |G|=1|G|=1.

And of course none of this really uses that KK is a field in the first place.
– Tobias Kildetoft
2 days ago

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@rschwieb: excellent, a simple and nice argument. Thanks for your time!
– user10
2 days ago

@user10 No problemo
– rschwieb
2 days ago