Iwasawa invariants and class numbers of quadratic fields for the prime 3

Let d be a square-free integer with d = 1 (mod 3) and d >0. Put k(+) = Q(root d) and k(-) = Q (root-3d). For the cyclotomic Z(3)-extension k(infinity)(+) of k(+), we denote by k(n)(+) the n-th layer of k(infinity)(+) over k(+). We prove that the 3-Sylow subgroup of the ideal class group of k(n)(+) is trivial for all integers n greater than or equal to 0 if and only if the class number of k(-) is not divisible by the prime 3. This enables us to show that there exist infinitely many real quadratic fields in which 3 splits and whose Iwasawa lambda(3)-invariant vanishes.