Unlimited lists of fundamental units of quadratic fields - Applications: Units of norm −1, Norm equations, Non-p -rational fields, p-Class groups)

26 pages. Elementary subject about real quadratic fields with new algorithms using given PARI programsWe use the polynomials m_s(t) = t^2 − 4s, s ∈ {−1, 1}, in an elementary process giving arbitrary large lists of fundamental units of quadratic fields of discriminants listed in ascending order. More precisely, let B ≫ 0; then as t grows from 1 to B, for each first occurrence of a square-free integer M ≥ 2, in the factorization m_s(t) =: M r^2 , the unit 1/2(t + r √ M) is the fundamental unit of norm s of Q(√ M), even if r > 1 (Theorem 4.6). Using m_sν (t) = t^2 − 4sν, ν ≥ 2, the algorithm gives arbitrary large lists of fundamental solutions to u^2 − M v^2 = 4sν (Theorem 4.11). We deduce, for p > 2 prime, arbitrary large lists of non p-rational quadratic fields (Theorems 6.3, 6.4, 6.5) and of degree p − 1 imaginary fields with non-trivial p-class group (Theorems 7.1, 7.2). PARI programs are given to be copied and pasted