Yesterday, at the last algebraic number theory lecture of the term, I defined the fundamental group \pi_1(O_K) of the ring of algebraic integers O_K in an algebraic number field K as the Galois group Gal(F/K), where F is the compositum of all the unramified extensions of K. And then, it was stated that the ideal class group Cl_K that we’ve been studying is just the abelianization of \pi_1(O_K). The point is that it seems reasonable now to start making computations of the whole \pi_1 rather than just the class group, if we want to capture finer invariants reflecting the complexity of a field. Of course, I’m not sure yet how to begin! But that’s what the best problems are like. Incidentally, the definition I gave isn’t quite right. One can give a better definition of \pi_1(O_k), closer to topological loops, after which the isomorphism with the Galois group becomes a theorem.
One of these days, I really will write some expository introduction at the undergraduate level. In the meanwhile the connection to topological fundamental groups is alluded to in the London-Paris lecture mentioned in an earlier post.
The basic reference in the field explaining this view is the great work Séminaire de Géométrie Algébrique, Vol. I by Alexandre Grothendieck and his students. Of course it’s hard to read if you’ve only now taken my course. But it’s good to get a sense of the classics even at this stage.
MK