We use the polynomials $m_s(t) = t^2 - 4 s$, $s \in \{-1, 1\}$, in an elementary process giving unlimited lists of {\it fundamental units of norm $s$}, of real quadratic fields, with ascending order of the discriminates. As $t$ grows from $1$ to an upper bound $\textbf{B}$, 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 4.2). Using $m_{s\nu}(t) = t^2 - 4 s \nu$, $\nu \geq 2$, the algorithm gives unlimited lists of {\it fundamental integers of norm $s\nu$} (Theorem~4.6). We deduce, for any prime $p>2$, unlimited lists of {\it non $p$-rational} quadratic fields (Theorems 6.3, 6.4, 6.5) and lists of degree $p-1$ imaginary fields with {\it non-trivial $p$-class group} (Theorems 7.1, 7.2). All PARI programs are given.
Fundamental units Norm equations PARI programs $p$-class numbers $p$-rationality Real quadratic fields
Birincil Dil | İngilizce |
---|---|
Konular | Temel Matematik (Diğer) |
Bölüm | Articles |
Yazarlar | |
Erken Görünüm Tarihi | 15 Eylül 2023 |
Yayımlanma Tarihi | 17 Eylül 2023 |
Gönderilme Tarihi | 14 Temmuz 2023 |
Kabul Tarihi | 14 Eylül 2023 |
Yayımlandığı Sayı | Yıl 2023 Cilt: 6 Sayı: 3 |
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