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

TL;DR

This paper presents an elementary method to generate arbitrarily large lists of fundamental units and solutions in quadratic fields, with applications to non p-rational fields and fields with non-trivial p-class groups.

Contribution

It introduces a novel elementary process using specific polynomials to systematically produce fundamental units and solutions in quadratic fields, expanding the known examples and applications.

Findings

Generates large lists of fundamental units for quadratic fields.

Produces arbitrarily large lists of solutions to certain Pell-type equations.

Identifies non p-rational quadratic fields and fields with non-trivial p-class groups.

Abstract

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.