Well, you adjoin all roots of unity to the field Q and obtain the field Qcycl. I claim that all Sylow subgroups S(l) of the absolute Galois group of the field Qcycl 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 Qcycl 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(GK,Z/l) over all finite extensions K of Q contained in K(l) (where GK 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(GK,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(GK,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 Kp of K (including the p=l and the real completions, of course). So I have to kill an element of Br(Kp) by adjoining roots of unity to Kp. But an element of Br(Kp) dies in a given extension of Kp 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 Kp roots of unity of orders lN with high enough N, one obtains extensions of Kp of degree divisible by arbitrarily high powers of l. But this is clear.
here is why
So it suffices to prove the following: for any finite extension K of Q and any element x in H2(GK,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(GK,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 Kp of K (including the p=l and the real completions, of course). So I have to kill an element of Br(Kp) by adjoining roots of unity to Kp. But an element of Br(Kp) dies in a given extension of Kp 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 Kp roots of unity of orders lN with high enough N, one obtains extensions of Kp of degree divisible by arbitrarily high powers of l. But this is clear.