(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If f(x) is an irreducible cubic polynomial over a field F, is it ever possible that [itex] C_2 [/itex] may occur as the [itex] \operatorname{Aut}(K/F) [/itex] where K is the splitting field of f?

3. The attempt at a solution

It seems that this should be theoretically possible. In particular, if f is an inseparable polynomial which split into precisely two distinct roots, this would be the case. However, I'm trying to think of an example and am having trouble coming up with one. In particular, it feels as though this won't be possible in characteristic 0, but may be possible in char p. Can anybody shed some light on this?

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# C2 as Galois group of an irreducible cubic

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