On the 3-Rank of the Class Group of Quadratic Fields

Let n 1, r 0 and s 0 be integers satisfying 4+r+3s 3 n+1 .Given linear polynomials f i (x) = m i x + n i for 1 i r + s, where the coefficients m i , n i are positive integers satisfying certain conditions, we prove that there exist infinitely many fundamental discriminants D > 0 such that the 3-rank of the class group of each quadratic fields) is simultaneously less than n.For a positive integer k, let g 1 , . . ., g k Qx be polynomials taking integer values at integers with g i (0) = 0. We also prove that there exist positive integers a, d such that each a + g i (d) is a fundamental discriminant and the 3-rank of the class group of each quadratic field) is simultaneously less than n.Moreover, these discriminants can be chosen all positive or all negative, giving either all real or all imaginary quadratic fields Q( a + g i (d)).