Unlimited lists of fundamental units of quadratic fields -- Applications to some arithmetic properties

Abstract

We use the polynomials $m_s(t)=t^2-4s$, $s\in \{-1,1\}$, 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 $M\ge 2$, in the factorization $m_s(t)=:M{r}^2$, the unit $\frac{1}{2}(t+r\sqrt{M})$ is the fundamental unit of norm $s$ of $\mathbb{Q}(\sqrt{M})$, even if $r>1$ (Theorem 4.1). Using $m_{s\nu }(t)=t^2-4s\nu$, $\nu \ge 2$, 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.

Minor corrections and new numerical results

References in corpus (2)

• Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields
• Weber's class number problem and $p$-rationality in the cyclotomic $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$