Let \(q\) be an odd prime power and \(D\) a monic squarefree polynomial in \(\mathbb F_q[t]\). The author proves that there are \(\gg q^{\ell/(2g)}\) such polynomials of even degree \(\leq \ell\) such that the class group of the real quadratic extension \(\mathbb F_q(t,\sqrt{D}\,)\) has an element of order \(g\). The corresponding result for number fields was proved by M. Ram Murty [Math. Appl., Dordr. 467, 229–239 (1999; Zbl 0993.11059)].