Inventing conjectures which I can't ever possibly prove

[posic]

is a strange activity, so why am I doing it? Well, back in 1994 I was more enthusiastic, that't one of the reasons.

Anyway, here is a conjecture: let F be a field. Let K be the field obtained by adjoining to F all roots of all orders of all elements of F. Then K is a field of homological dimension 1, that is, all Sylow subgroups of the absolute Galois group of K are free pro-l-groups.

Why do I believe it? Well, it is true for number fields (where it suffices to adjoin the roots of unity). And if F is Henzelian with respect to a discrete valuation and f is the residue field, then the conjecture is true for F whenever it holds for f. And this is about all the supporting evidence I have.

Comments

[french-man.livejournal.com]

Погодите, чего-то я не понимаю. Что такое order of an element of a field?

[posic.livejournal.com]

No, not an order of an element, but all roots of all orders. For each element a in F, you adjoin all roots of the element a. That is, adjoin to F all roots of the equations xⁿ-a=0, for all n, and all a in F.

When F contains all roots of unity, it will be just the maximal abelian extension of F.

[french-man.livejournal.com]

Тогда, извиняюсь, почему в случае числовых полей можно ограничится корнями из 1?

here is why

[posic.livejournal.com]

Well, you adjoin all roots of unity to the field Q and obtain the field Q^cycl. I claim that all Sylow subgroups S(l) of the absolute Galois group of the field Q^cycl are free pro-l-groups. It suffices to check that H²(S(l),Z/l)=0 for all l. Now let K(l) be the (infinite) extension of Q^cycl corresponding to the subgroup S(l). Lemma: the Galois cohomology commutes with inductive limits of fields. So H²(S(l),Z/l) is equal to the inductive limit of H²(G_K,Z/l) over all finite extensions K of Q contained in K(l) (where G_K denotes the absolute Galois group of K).

So it suffices to prove the following: for any finite extension K of Q and any element x in H²(G_K,Z/l) there exists a field L obtained by adjoining to K some root of unity such that the element x dies in the cohomology of L. We can assume that K contains the primitive root of unity of order l. Then H²(G_K,Z/l) is isomorphic to the kernel of multiplication with l in the Brauer group Br(K), and the same holds for any field containing K. So I have to prove that any element of Br(K) can be killed by adjoining some root of unity to K.

But this is elementary class field theory. The group Br(K) is embedded into direct sum of the Brauer groups of p-adic completions K_p of K (including the p=l and the real completions, of course). So I have to kill an element of Br(K_p) by adjoining roots of unity to K_p. But an element of Br(K_p) dies in a given extension of K_p if and only if the degree of the extension is divisible by the order of this element.

So it remains to notice that by adjoining to K_p roots of unity of orders l^N with high enough N, one obtains extensions of K_p of degree divisible by arbitrarily high powers of l. But this is clear.

Re: here is why

[french-man.livejournal.com]

Понял, спасибо.