1. Introduction. It is well known that there are infinitely many quadratic number fields and function fields with class number divisible by a given integer n (see Nagell [15] (1922) for imaginary number fields, Yamamoto [22] (1969) and Weinberger [21] (1973) for real number fields, and Friesen [6] (1990) for function fields). A related question concerns the n-rank of the field, that is, the greatest integer r for which the class group contains a subgroup isomorphic to (Z/nZ) r . In [22], Yamamoto showed that infinitely many imaginary quadratic number fields have n-rank at least 2 for any positive integer n ≥ 2. In 1978, Diaz y Diaz [3] developed an algorithm for generating imaginary quadratic fields with 3-rank 2, and Craig [2] showed in 1973 that there are infinitely many real quadratic number fields with 3-rank at least 2 and infinitely many imaginary quadratic number fields with 3-rank at least 3. A few examples of higher 3-rank have also been found (see for instance Llorente and Quer [14,18] who found three imaginary quadratic number fields with 3-rank 6 in 1987/1988). In a recent paper [4], Erickson, Kaplan, Mendoza, Shayler, and the author gave infinite, simply parameterized families of real and imaginary quadratic fields with 3-rank 2. Here we give a function field analogue.Note that Bauer, Jacobson, Lee, and Scheidler [1] have given algorithms which yield imaginary quadratic function fields with 3-rank at least 2 and a possibly empty set of imaginary quadratic function fields with 3-rank at least 3. The construction below yields infinitely many quadratic function fields, of any given signature, with 3-rank at least 2. See [9], [11], [12], [16], and [17] for constructions of function fields of arbitrary degree m with large n-rank for general n.Throughout we let q be a power of an odd prime, q ≡ 1 (mod 3). We use sgn(f ) to denote the leading coefficient of a polynomial f ∈ F q [T ], and we let |f | = q deg(f ) for f ∈ F q [T ]. The main result is as follows.