The Structure of the Class Groups of Global Function Fields with any Unit Rank

construct, for positive integers m, n and r with 0 ≤ r ≤ m − 1, infinitely many global function fieldsK of degreemover F(T ) such thatK has unit rank r and the ideal class group of K contains a subgroup isomorphic to (Z/nZ)m−r . This work improves Pacelli’s work [11] by increasing the subgroup rank from m − r − 1 to m − r, and also we obtain the result for other behavior of the prime at infinity. In particular, for the unit rank r = 0, in [7] we worked on the case in which the infinite prime is inert, so in this paper we complete the case in which the infinite prime is totally ramified. For 0 ≤ r ≤ m − 1, we consider the following two cases for the splitting behavior of the prime at infinity: The prime at infinity splits into r + 1 primes, one with relative degree m − r, the others with relative degree 1, all unramified, or the prime at infinity splits into r + 1 primes, one with ramification index m − r, the others unramified, all with relative degree 1.