Unlimited Lists of Quadratic Integers of Given Norm Application to Some Arithmetic Properties

Abstract

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 of norm $s$},
 of real quadratic fields, with ascending order of the discriminants. 
 As $t$ grows from $1$ to an upper bound $\BB$, 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 $\Q(\sqrt M)$, even if $r >1$ (Theorem \ref{fondunit}).
 Using $m_{s\nu}(t) = t^2 - 4 s \nu$, $\nu \geq 2$, the algorithm gives unlimited 
 lists of {\it fundamental integers of norm $s\nu$} (Theorem~\ref{unicity}). 
 We deduce, for any prime $p>2$, unlimited lists of {\it non $p$-rational} 
 quadratic fields (Theorems \ref{prational}, \ref{heuristic}, \ref{infty}) and lists 
 of degree $p-1$ imaginary fields with non-trivial $p$-class group (Theorems 
 \ref{cubic}, \ref{quintic}). All PARI programs are given.