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

• URL: http://arxiv.org/abs/2406.14632v1
• Date: Thu, 20 Jun 2024 18:00:51 GMT
• Title: Some new infinite families of non-$p$-rational real quadratic fields
• Authors: Gary McConnell,
• Abstract summary: We give a simple methodology for constructing an infinite family of simultaneously non-$p_j$-rational real fields, unramified above any of the $p_j$. One feature of these techniques is that they may be used to yield fields $K=mathbbQ(sqrtD)$ 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
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Fix a finite collection of primes $\{ p_j \}$, 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-$p_j$-rational real quadratic fields, unramified above any of the $p_j$. Alternatively these may be described as infinite sequences of instances of $\mathbb{Q}(\sqrt{D})$, for varying $D$, where every $p_j$ 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 $K=\mathbb{Q}(\sqrt{D})$ 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 $p^a$ for $a\geq1$ arbitrarily large.

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