Let F be a finite field with q elements, and T a transcendental element over F: In this paper, we construct infinitely many real function fields of any fixed degree over FðTÞ with ideal class numbers divisible by any given positive integer greater than 1. For imaginary function fields, we obtain a stronger result which shows that for any relatively prime integers m and n with m; n41 and relatively prime to the characteristic of F; there are infinitely many imaginary fields of fixed degree m such that the class group contains a subgroup isomorphic to ðZ=nZÞ mÀ1 : r 2004 Elsevier Inc. All rights reserved.

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In this paper we construct, for any integers m and n, and 2 g m − 1, infinitely many function fields K of degree m over F(T ) such that the prime at infinity splits into exactly g primes in K and the ideal class group of K contains a subgroup isomorphic to (Z/nZ) m−g . This extends previous results of the author and Lee for the cases g = 1 and g = m.

1. Introduction. It is well known that there are infinitely many quadratic number fields with class number divisible by a given integer n (see Nagell [8] (1922) for imaginary fields and Yamamoto [11] (1970) and Weinberger [10] (1973) for real 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 [11], Yamamoto showed that infinitely many imaginary quadratic number fields have n-rank ≥ 2 for any positive integer n ≥ 2. In 1978, Diaz y Diaz [2] developed an algorithm for generating imaginary quadratic fields with 3-rank at least 2, and Craig [1] 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 [6,9] who found in 1987/1988 three imaginary quadratic number fields with 3-rank 6). In this paper, we give infinite, simply parameterized families of real and imaginary quadratic fields with 3-rank 2. Although the existence of such fields has been known, the fields here are much easier to describe, and the parameterization yields a new lower bound on the number of fields with discriminant < x and 3-rank ≥ 2 (see [7]).The main result is as follows:

Let F be a finite field and T a transcendental element over F. In this paper, we construct, for integers m and n relatively prime to the characteristic of F(T ), infinitely many imaginary function fields K of degree m over F(T ) whose class groups contain subgroups isomorphic to (Z/nZ) m . This increases the previous rank of m − 1 found by the authors in [Y. Lee, A. Pacelli, Class groups of imaginary function fields: The inert case, Proc. Amer. Math. Soc. 133 (2005) 2883-2889].