  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.
