Unlimited lists of fundamental units of quadratic fields -- Applications to some arithmetic properties
| Autor: | Gras, Georges |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Communications in Advanced Mathematical Sciences Vol. 6, No. 3, 148-176, 2023 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.33434/cams.1327372 |
| Popis: | We use the polynomials $m_s(t) = t^2 - 4 s$, $s \in \{-1, 1\}$, in an elementary process giving arbitrary large lists of {\it fundamental units} of quadratic fields of discriminants listed in ascending order. More precisely, let $\mathbf{B} \gg 0$; then as $t$ grows from $1$ to $\mathbf{B}$, for each {\it first occurrence} of a square-free integer $M \geq 2$, in the factorization $m_s(t) =: M r^2$, the unit $\frac{1}{2} \big(t + r \sqrt{M}\big)$ is the fundamental unit of norm $s$ of $\mathbb{Q}(\sqrt M)$, even if $r >1$ (Theorem 4.1). Using $m_{s\nu}(t) = t^2 - 4 s \nu$, $\nu \geq 2$, the algorithm gives arbitrary large lists of {\it fundamental solutions} to $u^2 - M v^2= 4s\nu$ (Theorem 4.11). We deduce, for $p>2$ prime, arbitrary large lists of {\it non $p$-rational} quadratic fields (Theorems 6.3, 6.4, 6.5) and of degree $p-1$ imaginary fields with non-trivial $p$-class group (Theorems 7.1,7.2). PARI programs are given to be copied and pasted. Comment: Minor corrections and new numerical results |
| Databáze: | arXiv |
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