Mixiu

    I will talk about 'étale fundamental group' , which Grothendieck defines on a scheme . This is analogy of the fundamental group of a topological space and  a generalization of the Galois group of a field. 

Section 1 

    We started with a topological space. For a good topological space X , we can defines the fundamental group π_1(X,x) as the homotopy class of loops start from a given point x , and there is a theorem says:" the category of covering space of X is equivalence to the category of π_1(X,x) sets"  

     In number theory, there is  Galios theorem  which is very similar to the previous theorem . Pick a field k , let G be the Galois group of k^(s)/k, the automorphism group of k^(s) that fixes k, here k^(s) is the separable closure of k. Galois theorem says there is a correspondence between the open subgroup of the G and the finite separable extension of k. 

   We formulate is in a categorical way: From a open subgroup H of G, we can get a continuous action of the group G on the