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Could I have an example of a finite etale group scheme over a field k which is not a constant group scheme?

I just know that the category of etale group schemes over a k is equivalent to the category of abstract groups with a galois action. So I am wondering what the objects would be like in the left category. In fact any finite etale scheme over k should be a finite union of spectrums of fields which are finite separable extensions of k, and if it is a group scheme then it should have a rational point, but I don't see how one can translate the rational point to other points, so I guess there might be some non-rational point in the group scheme. is that right? an example?

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    $\begingroup$ How about $\mu_n$, the $n$-th roots of $1$, over a field $k$ in which $n$ is invertible but which does not contain a primitive $n$-root of $1$ ? $\endgroup$
    – Chandan Singh Dalawat
    Commented Nov 4, 2012 at 15:27
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    $\begingroup$ For $k=\mathbf{Q}$, it works for every $n>2$. $\endgroup$
    – Chandan Singh Dalawat
    Commented Nov 4, 2012 at 16:08
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    $\begingroup$ Yes. Let $x$ be an element which is not an $n$th power in $k$. Then the set of $n$th roots of $x$ is a torsor for the $n$th roots of unity in an obvious way, and has no $k$-valued points, thus is nontrivial. $\endgroup$
    – Will Sawin
    Commented Nov 4, 2012 at 16:25
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    $\begingroup$ If you have a surjective morphism of commutative groups $G\to H$ with kernel $N$, then every fibre is a torsor under $N$. $\endgroup$
    – Chandan Singh Dalawat
    Commented Nov 4, 2012 at 16:25
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    $\begingroup$ @stefan: every torsor under an étale group scheme is étale as a scheme (or algebraic space), because étaleness is an fppf-local property. Hence it is trivialized by an étale covering (namely itself). $\endgroup$
    – Laurent Moret-Bailly
    Commented Nov 5, 2012 at 7:19

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