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 : 58 | Download : 28

Publication Date : 2023-09-17

Article Type : Research Paper

Abstract :We use the polynomials $m_sinsert ignore into journalissuearticles values(t); = t^2 - 4 s$, $s \\in \\{-1, 1\\}$, 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 $m_sinsert ignore into journalissuearticles values(t); =: M r^2$, the unit $\\frac{1}{2} \\biginsert ignore into journalissuearticles values(t + r \\sqrt{M}\\big);$ is the fundamental unit of norm $s$ of $\\mathbb{Q}insert ignore into journalissuearticles values(\\sqrt M);$, even if $r >1$ insert ignore into journalissuearticles values(Theorem 4.2);. Using $m_{s\\nu}insert ignore into journalissuearticles values(t); = t^2 - 4 s \\nu$, $\\nu \\geq 2$, the algorithm gives unlimited lists of {\\it fundamental integers of norm $s\\nu$} insert ignore into journalissuearticles values(Theorem~4.6);. We deduce, for any prime $p>2$, unlimited lists of {\\it non $p$-rational} quadratic fields insert ignore into journalissuearticles values(Theorems 6.3, 6.4, 6.5); and lists of degree $p-1$ imaginary fields with {\\it non-trivial $p$-class group} insert ignore into journalissuearticles values(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

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