论文标题
二次字段的基本单元的无限列表 - 某些算术属性的应用
Unlimited lists of fundamental units of quadratic fields -- Applications to some arithmetic properties
论文作者
论文摘要
我们在基本过程中使用多项式$ m_s（t）= t^2-4 s $，$ s \ in \ { - 1，1，1，1，1 \} $，给出了{\ it基本单位}的任意大列表，该{\ it基本单位}的二次判别字段列出了升级顺序。更准确地说，令$ \ mathbf {b} \ gg 0 $;然后随着$ t $从$ 1 $增长到$ \ mathbf {b} $，对于每个{\ iT首次出现}即使$ r> 1 $（Theorem 4.1），$ \ mathbb {q} $ \ mathbb {q}的基本单位（\ sqrt m）$也是如此。使用$ m_ {sν}（t）= t^2-4sν$，$ν\ geq 2 $，该算法将{\ it基本解决方案}的任意大列表提供给$ u^2 -m v^2 =4Sν$（theorem 4.11）。我们以$ p> 2 $ prime的形式推断出{\ it non $ p $ - 理性}二次字段（定理6.3、6.4、6.5）和$ p-1 $ p-1 $假想领域的任意大列表（定理$ p $ p $ -Class Group）（定理7.1,1.1,7.2）。 Pari计划被复制和粘贴。
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 $m_{sν}(t) = t^2 - 4 s ν$, $ν\geq 2$, the algorithm gives arbitrary large lists of {\it fundamental solutions} to $u^2 - M v^2= 4sν$ (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.
