PRIMITIVE RECURSIVE DECIDABILITY FOR LARGE RINGS OF ALGEBRAIC INTEGERS INSIDE THE COMPOSITUM OF ALL SYMMETRIC EXTENSIONS OF Q

Let Q(symm) be the compositum of all finite Galois extensions of Q with symmetric Galois groups. Denote the absolute Galois group of Q by Gal(Q). For each sigma = (sigma(1),..., sigma(e)) is an element of Gal(Q)(e), let Q(symm)(sigma) be the subextension of Q(symm)/Q fixed by sigma(1),..., sigma(e) and let Q(symm)[sigma] be the maximal Galois extension of Q inside Qsymm(sigma). Also, let Z(symm)(sigma) (resp. Z(symm)[sigma]) be the ring of integers inside Q(symm)(sigma) (resp. Q(symm)[sigma]). Then, the theory of all sentences in the language of rings which are true in Q(symm)(sigma) (resp. Q(symm)[sigma], Z(symm)(sigma), Z(symm)[sigma]) for almost all (with respect to the Haar measure) sigma is an element of Gal(Q)(e) is primitive recursively decidable.