On the fundamental units and the class numbers of real quadratic fields

Source abstract

Let Q be the rational number field and h(m ) be the class number of the real quadratic field with a positive square-free integer m . It is known that if h(m ) = 1 holds, then m is one of the following four types with prime numbers p ≡ 1, p t ≡ 3 (mod 4) (1 昤 i ≥ 4) : i) m = p, ii) m = p 1 , iii) m = 2 or m = 2p 2 , iv) m = p 3 p 4 (see Behrbohm and Rédei [1]). The sufficient conditions for h(m ) > 1 with these m were given by several authors: in all cases by Hasse [2], in case i) by Ankeny, Chowla and Hasse [3] and by Lang [4], in case ii) by Takeuchi [5] and by Yokoi [6].