Etale algebras over dual numbers How to classify all etale algebras over $k[x]/x^2$ where $k$ is some field? All etale covers? What is the classification for an arbitrary Artin algebra?
My attempt: Let's first assume that $k$ is algebraically closed. Let $k[x]/x^2 \to A$ be an etale map. Then $k \to A \otimes_{k[x]/x^2} k$ is etale. Therefore, $A/x = \Pi_{i \in I} k$ where $|I|$ is the degree of $A$. Does it follow that $A = \Pi_{i \in I} k[x]/x^2$? I'm not sure about that.
 A: One can just do this explicitly, but I thought I'd mention that there is a nice aspect of the theory that makes the comments from above precise. The key thing that makes everything nice is the topological invariance of the etale site:

Theorem(Topological invariance of the etale site, Tag 04DZ): *Let $f\colon X\to Y$ be a universal homeomorphism. Then, the pullback functor $(V\to Y)\mapsto (V\times_Y X\to X)$ defines an equivalence of categories
$$\left\{\begin{matrix}\text{Schemes etale}\\ \text{over }Y\end{matrix}\right\}\to \left\{\begin{matrix}\text{Schemes etale}\\ \text{over }X\end{matrix}\right\}.$$

In particular, as $\mathrm{Spec}(k)\to \mathrm{Spec}(k[x]/(x^2))$ is a universal homeomorphism (being a nilpotent closed immersion) we see that the functor
$$\left\{\begin{matrix}\text{Schemes etale}\\ \text{over }\mathrm{Spec}(k[x]/(x^2))\end{matrix}\right\}\to \left\{\begin{matrix}\text{Schemes etale}\\ \text{over }\mathrm{Spec}(k)\end{matrix}\right\}.$$
is an equivalence. Now, as you are certainly well-aware, the etale algebras over $\mathrm{Spec}(k)$ are of the form $\bigsqcup_i\mathrm{Spec}(L_i)$ with $L_i/k$ a finite etale extension. As $\bigsqcup_i \mathrm{Spec}(L_i[x]/(x^2))$ pulls back along $\mathrm{Spec}(k)\to \mathrm{Spec}(k[x]/(x^2))$ to $\bigsqcup_i\mathrm{Spec}(L_i)$ we deduce that, up to isomorphism, every etale scheme over $\mathrm{Spec}(k[x]/(x^2))$ is isomorphic to such a $\bigsqcup_i \mathrm{Spec}(L_i[x]/(x^2))$.
