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

Number Theory 2024-06-24 v1 Quantum Physics

Abstract

Fix a finite collection of primes {pj}\{ p_j \}, not containing 22 or 33. 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-pjp_j-rational real quadratic fields, unramified above any of the pjp_j. Alternatively these may be described as infinite sequences of instances of Q(D)\mathbb{Q}(\sqrt{D}), for varying DD, where every pjp_j is a kk-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)K=\mathbb{Q}(\sqrt{D}) for which a pp-power cyclic component of the torsion group of the Galois groups of the maximal abelian pro-pp-extension of KK unramified outside primes above pp, is of size pap^a for a1a\geq1 arbitrarily large.

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Cite

@article{arxiv.2406.14632,
  title  = {Some new infinite families of non-$p$-rational real quadratic fields},
  author = {Gary McConnell},
  journal= {arXiv preprint arXiv:2406.14632},
  year   = {2024}
}

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11 pages