I apologize if the question is bit trivial for mathoverflow, but I asked on stack exchange a while ago and haven't got any answer.

Let $k$ be a field of characteristic $p > 0$. Consider the multiplicative group scheme $\mathbb{G}_m$ and its subgroup $\mu_p$, the $p$-th roots of unity. It is well known that the quotient presheaf $\mathbb{G}_m / \mu_p$ is not a sheaf in fppf topology, and its sheafification in fppf topology is representable by $\mathbb{G}_m$ via the morphism $\mathbb{G}_m \xrightarrow{\cdot \mapsto \cdot^p} \mathbb{G}_m$. However, is the quotient presheaf $\mathbb{G}_m / \mu_p$ an **étale** sheaf on the category of $k$-schemes?

If $U \rightarrow X$ is an étale cover, I can prove the equalizer sequence $\mathbb{G}_m / \mu_p(X) \rightarrow \mathbb{G}_m / \mu_p (U) \rightarrow \mathbb{G}_m / \mu_p(U\times_X U)$ is exact when X is **reduced**. In fact, if $s \in \mathcal{O}_X(U)^\times$ and $(\frac{s\otimes 1}{1\otimes s})^p = 1$, since $U\times_X U$ is again reduced we can deduce $s\otimes 1 = 1\otimes s$ hence $s\in \mathcal{O}_X(X)^\times$. However I don't know to prove the general case when $X$ is not reduced. I tried several ways to formulate an induction, but couldn't work it out.

Appreciate any hints towards a proof or a counterproof. Thank you!

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