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I have a question about a claim about classification of finite flat group schemes: https://en.wikipedia.org/wiki/Group_scheme#Finite_flat_group_schemes

A fin flat group scheme $G$ is of type $(a,b)$; $a,b \in \{\text{ etale, connected }\}$ if $G$ is $a$ and it's Cartier dual (https://en.wikipedia.org/wiki/Cartier_duality) $ G^{\vee}$ is $b$.

A well know statement says that is $G$ is a fin flat group scheme over a field $k$ of characteristic $0$ is always etale. (*)

Essentially the argument works as follows: since $G$ is Artin ring, it is a finiteproduct of local rings of dimension zero. Then we can apply the Theorem that every fin flat group scheme over a field $k$ of characteristic $0$ is always reduced, and therefore every local ring in the product is a field, the proof of the used Theorem can be found for example here: https://www.uni-frankfurt.de/52288632/Stix_finflat_Grpschemes.pdf or here:https://math.dartmouth.edu/~jvoight/notes/274-Schoof.pdf

Question: Listening a lecture my prof mentioned that the statement (*) follows also easily from the well known algebra fact that every irreducible polynomial over field of characeristic zero is separable.

What I not understand is why this elemenary fact also imply statement (*) immediately? I don't see the connection. Does anybody have an idea what my prof had with this comment in mind?

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    $\begingroup$ Have you thought about asking your professor? $\endgroup$
    – S. Carnahan ♦
    Commented Dec 1, 2019 at 16:08
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    $\begingroup$ Most probably he/she just meant that this elementary fact (together with the fact that in characteristic 0 every finite extension is generated by a single element) implies that the spectrum of a finite field extension in characteristic 0 is etale. Still you need to use Cartier theorem to show that any group scheme is reduced. $\endgroup$
    – Victor Petrov
    Commented Dec 1, 2019 at 16:58
  • $\begingroup$ @S.Carnahan: I did, it was suggested to think a while about it :) But yes, it seems that the argument was exactly that one Victor Pertrov explained above. $\endgroup$
    – user267839
    Commented Dec 1, 2019 at 21:28

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