C2 as Galois group of an irreducible cubic

[Kreizhn]

Homework Statement

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

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?

 

[morphism]

If your irreducible cubic f(x) is separable, then its Galois group will either be A_3 or S_3. If f(x) isn't separable, then necessarily charF=3, and f(x)=x^3-a for some noncube a. Can C_2 be the splitting field of such an f(x)?

 

[Kreizhn]

Ah yes. For some reason it didn't occur to me that the field would necessarily have to have characteristic 3, but that makes sense. In that case, the polynomial is purely inseparable and so $\operatorname{Aut}(K/F)$ would be trivial.