Construction of certain real quadratic fields
Tsuyoshi Uehara · Proceedings of the Japan Academy Series A Mathematical Sciences · 1983
Let n be a given natural number.In this note we shall construct real quadratic fields whose undamental units are congruent to _+1 modulo n.We also give a new proof of the existence o]? infinitely many real quadratic fields each with class number divisible by n (cf.Let Z, Q be the ring of rational integers, the field of rational numbers respectively.For a rational integer m=/=0 and a prime p we denote by ord m the greatest nonnegative rational integer f such that m----0 (mod p).Lemma.Let , be integers of a quadratic field K such that a--+n for somen>l inZ.We write=(a+b/ d)/2, /=(s+t,/ d)/2 with a, b, s, t in Z, where d is the discriminant of K.If p is a prime dividing d such that ord, a=ord, 2, then we have ord, t ord b-ordp n except in the following two cases" (i) p=2, ord2 d=2 and n--O (mod 2), (ii) p=3, d=_6 (mod 9) and n-O (mod 3).