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 discriminates. As $t$ grows from $1$ to an upper bound $\textbf{B}$, 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 4.2). 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~4.6). We deduce, for any prime $p>2$, unlimited lists of {\it non $p$-rational} quadratic fields (Theorems 6.3, 6.4, 6.5) and lists of degree $p-1$ imaginary fields with {\it non-trivial $p$-class group} (Theorems 7.1, 7.2). All PARI programs are given.

Keywords

• fundamental units
• norm equations
• pari programs
• $p$-class numbers
• $p$-rationality
• real quadratic fields