UNLIMITED LISTS OF QUADRATIC INTEGERS OF GIVEN NORM APPLICATION TO SOME ARITHMETIC PROPERTIES

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 norm $s$},
 real quadratic fields, with ascending order discriminants. As $t$ grows from $1$ to upper bound $\BB$, for each first occurrence} a square-free integer $M \geq 2$, factorization =: M r^2$, the
 unit $\frac{1}{2} \big(t + r \sqrt{M}\big)$ is $s$
 $\Q(\sqrt M)$, even if $r >1$ (Theorem \ref{fondunit}).
 Using $m_{s\nu}(t) s \nu$, $\nu algorithm gives integers $s\nu$} (Theorem~\ref{unicity}). deduce, any prime $p>2$, non $p$-rational} fields (Theorems \ref{prational}, \ref{heuristic}, \ref{infty}) and degree $p-1$ imaginary non-trivial $p$-class group \ref{cubic}, \ref{quintic}). All PARI programs are given.