Local-global principles for rational points have a long and influential role in arithmetic geometry. Our focus will be on what a universal local-global principle for rational points implies about the Galois theory of the field, especially fields of totally S-adic numbers, where S is an infinite set of places. Some applications of our work include: (i) new evidence for the Shafarevich Conjecture on the freeness of the absolute Galois group of the maximal cyclotomic extension of k; (ii) "independence" of the decomposition groups above infinite families of primes. Time permitting, we will also mention several areas for future work.