@Chris'ssis I have to think about it for a while :) .. but +1 to the double gamma function .. I understand it'll be ultimately at the heart of any real method that we might come up with :)
@Alizter I think you give up too easily, you need some more practice. I saw it the last time I talked to you: you said the series you got is hard and didn't continue the work.
Anybody seeming broken notifications? Someone tagged me in a comment on a question, and now every page I go to says I have a new comment, even after I've read the comment.
@Chris'ssis I see that you put 'let's brilliantly compute .. ' before your proofs :) .. you can try 'let's smoothly compute ..' for the smooth one .. for a change :D
@Alyosha The natural homomorphism $\phi:G\to G/H$ means that for any subgroup $N$ of $G/H$ we can find the "inverse image" $\phi^{-1}(N)=\{g\in G: \phi(g)\in N\}$, which can be proved to be a subgroup of $G$.
@Alyosha I won't do that if I were you. From where have you studied it?
@Alizter Oh, by the way, your problem seems interesting. After a few hours of fiddling, I think deciding whether or not every group realizable as a galois group of some algebraic number field can be realized as a galois group of some real algebraic number field is a highly nontrivial problem.
Can you draw me the Riemannsurface of $w = \sqrt[3]{z(z-1)}$?
@Alyosha You won't need much covering space theory while doing Alekseev. Just the intro.
Every Riemannsurface is more or less an "algebraic" branched covering of $\Bbb C$. In particular, this sense can be generalized, along with the concept of monodromy.
You wouldn't want to care for the Riemannsurface, really. This sense of group can be attached to any covering map from a topological space to another topological space.
This is the so-called "geometric galois action" that I yet have to study.
Thanks @Daniel. Oddly, the comment notification stopped when I got some rep points, but the rep points notification won't go away now. :( - You must login to post
@Alyosha There is no general algorithm! For example, I saw something a few days ago that has an infinite-genus Riemannsurface. Devil knows how to draw them.
The only rigorous definition I know comes from covering maps.
@Alyosha I haven't studied manifolds, sorry.
@Alyosha Have you ever thought of mimicking Galois theory in the sense of groups? We have pretty much all that we want for mimicking : a structure, a sense of inclusion, automorphisms.
I think I can prove that if N is a characteristic subgroup of H a characteristic subgroup of G then there is a short exact sequence 1 --> Aut_H(G) --> Aut_N(G) --> Aut_N(H) --> 1
The exactness probably follows by noting that since H char G, Aut(G) --> Aut(H) induced from restriction automatically gives the map Aut_N(G) --> Aut_N(H) which is clearly of kernel Aut_H(G), as Aut_H(G) is sitting inside Aut_N(G)
I am hesitating with the shortness. I think it's just left-exact.
@BalarkaSen to better understand my thoughts on "beauty of mathematics", you look at $$\sum_{n=1}^{\infty} \left(\psi^{(0)}(n)+\frac{1}{2}\psi^{(1)}(n)-\log(n)\right)$$ and know the answer. No need of pen and paper.
I didn't invent a special way here, but I learned a lot, and learning a lot I'm able now to produce such beautiful connections that lead to an amazing proof, you know.
This is the beauty of mathematics, not the stuff for kids. The stuff for kids is beautiful for kids.
