It is known, that the étale fundamental group of a normal connected scheme equals the galois group of the maximal unramified extension of its function field.


This is not true for integral schemes in general. The point is somehow, that one cannot construct flat finite covers from a function field extension as one does with normalization. So my questions are as follows:


1. What are examples for integral schemes, where the fundamental group is not a galois group of the function field? Or equivalently: A scheme with non-isomorphic étale covers but isomorphic function fields?
2. Is there already an example which is one-dimensional, e.g. an algebraic curve or an order in a number field? Somehow the étale covers should look different over the singularities. What can happen there?
3. In the case of curves it is known how to construct singular curves from normal ones. Is there a way of constructing étale covers of a singular curve by taking its normalization in a function field extension and then singularizing it again?


I thank you very much for your answers.



The étale fundamental group in the non-normal case
Take a nodal cubic $C$, that is a projective line of which you identify two points $0$ and $\infty$.
This curve has connected étale covers of any degree: take $n$ copies of the projective line,
numbered circularly, and identify the $\infty$ of each of them with the $0$ of the next one.
This is a $\mathbf Z/n\mathbf Z$-Galois cover of $C$.


(Taken from Hartsorne's *Algebraic Geometry*, page 276.)


