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| Autori principali: | , |
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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2408.03836 |
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| _version_ | 1866916349914120192 |
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| author | Qi, Peikai Stokes, Matt |
| author_facet | Qi, Peikai Stokes, Matt |
| contents | 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 $K$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Non p-Rationality and Iwasawa Invariants of Certain Real Quadratic Fields Qi, Peikai Stokes, Matt Number Theory 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 $K$. |
| title | On the Non p-Rationality and Iwasawa Invariants of Certain Real Quadratic Fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.03836 |