Salvato in:
| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2408.03836 |
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Sommario:
- Let $p$ be an odd prime, and $m,r \in \mathbb{Z}^+$ with $m$ coprime to $p$. In this paper we investigate the real quadratic fields $K = \mathbb{Q}(\sqrt{m^2p^{2r} + 1})$. We first show that for $m < C$, where constant $C$ depends on $p$, the fundamental unit $\varepsilon$ of $K$ satisfies the congruence $\varepsilon^{p-1} \equiv 1 \mod{p^2}$, which implies that $K$ is a non $p$-rational field. Varying $r$ then gives an infinite family of non $p$-rational fields. When $m = 1$ and $p$ is a non-Wieferich prime, we use a criterion of Fukuda and Komatsu to show that if $p$ does not divide the class number of $K$, then the Iwasawa invariants for cyclotomic $\mathbb{Z}_p$-extension of $K$ vanish. We conjecture that there are infinitely many $r$ such that $p$ does not divide the class number of $K$.