- Communications in Advanced Mathematical Sciences
- Cilt: 6 Sayı: 3
- Unlimited Lists of Quadratic Integers of Given Norm - Application to Some Arithmetic Properties
Unlimited Lists of Quadratic Integers of Given Norm - Application to Some Arithmetic Properties
Authors : Georges Gras
Pages : 148-176
Doi:10.33434/cams.1327372
View : 23 | Download : 25
Publication Date : 2023-09-17
Article Type : Research
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