Some new infinite families of non-p-rational real quadratic fields

Fix a finite collection of primes \ pⱼ \, not containing $2$ or $3$. Using some observations which arose from attempts to solve the SIC-POVMs problem in quantum information, we give a simple methodology for constructing an infinite family of simultaneously non-pⱼ-rational real quadratic fields, unramified above any of the pⱼ. Alternatively these may be described as infinite sequences of instances of Q(√D), for varying D, where every pⱼ is a k-Wall-Sun-Sun prime, or equivalently a generalised Fibonacci-Wieferich prime. One feature of these techniques is that they may be used to yield fields K=Q(√D) for which a p-power cyclic component of the torsion group of the Galois groups of the maximal abelian pro-p-extension of K unramified outside primes above p, is of size pᵃ for a≥1 arbitrarily large.