Do sporadic fields exist?

Asked Jul 18 '18 at 03:24

Active Jul 18 '18 at 03:30

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If we think of fields as groups that also have addition as well as multiplication. Then in the classification of simple groups there are a set of sporadic groups. Are there also such a thing as sporadic fields?

Or if not, are there at least some kind of exceptional fields that have unusual and unique structure? I guess, the complex and quaternions are fairly "unique".

asked Jul 18 '18 at 03:24

zooby

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Re your last sentence, I'd say $\Bbb R$ is "more unique" than $\Bbb C$ – Hagen von Eitzen Jul 18 '18 at 03:32
LMFDB lists some with weird properties – Ed Pegg Jul 18 '18 at 03:55
Every field contains either $\mathbb{Q}$, or $\mathbb{F}_{p}$ ($p$ prime) as a base field. I don't know if that helps answer your question, I'm not sure what would make a field "exceptional" in the context of this question. – xxxxxxxxx Jul 21 '18 at 00:33
Yeah perhaps fields are too restrictive to be interesting. Maybe I was thinking of algebras in general. – zooby Jul 21 '18 at 03:28
The quaternions aren't even a field. – Kenny Lau Jul 24 '18 at 18:08
@Kenny you're right. – zooby Jul 24 '18 at 20:50

Do sporadic fields exist?

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