Let $A \to B$ be a ring extension.


What is the definition of $B/A$ étale ?


When $A$ is a field, do we get a nice characterization ?



Definition of étale for rings
You say that a ring homomorphism $\phi: A \to B$ is **étale** (resp. **smooth**, **unramified**), or that $B$ is étale (resp. smooth, unramified) over $A$ is the following two conditions are satisfied:


* $A \to B$ is **formally étale** (resp. **formally smooth**, **formally unramified**): for every square-zero extension of $A$-algebras $R' \to R$ (meaning that the kernel $I$ satisfies $I^2 = 0$) the natural map $$\mathrm{Hom}\_A(B, R') \to \mathrm{Hom}\_{A}(B, R)$$ is bijective (resp. surjective, injective).
* $B$ is **essentially of finite presentation** over $A$: $A \to B$ factors as $A \to C \to B$, where $A \to C$ is of finite presentation and $C \to B$ is $C$-isomorphic to a localization morphism $C \to S^{-1}C$ for some multiplicatively closed subset $S \subset C$.


The second condition is just a finiteness condition; the meat of the concept is in the first one. Formal smoothness is often referred to as the infinitesimal lifting property. Geometrically speaking, it says that if the affine scheme $\mathrm{Spec} \ B$ is smooth over $\mathrm{Spec} \ A$, then any map from $\mathrm{Spec} \ R$ to $\mathrm{Spec} \ B$ lifts to any square-zero (and hence any infinitesimal) deformation $\mathrm{Spec} \ R'$. Moreover, if $\mathrm{Spec} \ B$ is étale over $\mathrm{Spec} \ A$ this lifting is unique.


Differential-geometrically, unramifiedness, smoothness and étaleness correspond to the tangent map of $\mathrm{Spec} \ \phi$ being injective, surjective and bijective, respectively. In particular, étale is the generalization to the algebraic case of the concept of local isomorphism.


There are two references you might want to consult. The first one, in which you can read all about the formal properties of these morphisms, is Iversen's "[Generic Local Structure of the Morphisms in Commutative Algebra](https://doi.org/10.1007/BFb0060790)" (Springer LNM 310). The second one, Hartshorne's book "[Deformation Theory](https://doi.org/10.1007/978-1-4419-1596-2)" (Graduate Texts in Mathematics 257), will give you a lot of information about the geometry; section 4 of chapter 1 talks about the infinitesimal lifting property.


EDIT: The [EGA](https://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique) definition of étale morphism of rings is slightly different from the above, in the sense that it requires finite presentation, not just locally of finite presentation: see the comments below.


