Reference for definition and more of Galois covering I encountered the term "galois covering" in Beauville's book on algebraic surfaces, as well as in the article "rational surfaces with many nodes" by Dolgachev et al.
However, i have not yet found a reference with the definition and some elaboration on the concept. Does anyone know a good one? (it is not in Hartshorne, right?)
Thanks!
 A: The laziest solution to the problem of defining (Galois) coverings in algebraic geometry  would  be to  copy verbatim the definition in topology, just replacing words like "topological space" by "algebraic variety".
However this doesn't work at all!
Already the simplest example $q:\mathbb C^*\to\mathbb C^*:z\mapsto z^2$ is not locally trivial downstairs in the Zariski topology.
This just means that if you delete a finite set $F$ from the copy of $\mathbb C^*$ downstairs, obtaining $U=\mathbb C^*\setminus F$, the inverse image $q^{-1}(U)$ is not a product $U\times D$  ($D=$ the variety with two elements).    
So  algebraic geometers have  decided that their   coverings would  just be finite  surjective morphism $p:Y\to X$ of algebraic varieties.
For example the above map $q$ is a covering by this definition, but so is also $Q:\mathbb C\to\mathbb C:z\mapsto z^2$. This is amazing because $Q$ is not  a topological covering.
If you want to say closer to topology, you introduce étale coverings, which are non-ramified and flat coverings.   
Coming back to the  general  definition, note that   a covering $p:Y\to X$ between irreducible varieties induces a finite field extension $p^*:k(X)\to k(Y)$ of their function fields.
This finally allows to answer  your question  :  
Definition  A Galois covering is a covering $p:Y\to X$ between irreducible  normal varieties for which the induced field extension $p^*:k(X)\to k(Y)$ is Galois.
A: A gentle introduction is Lenstra's Galois theory for schemes, online. The original reference is SGA1.
