The Stacks project

Exercise 111.22.5. Show that the class group of the ring $A = k[x, y]/(y^2 - f(x))$ where $k$ is a field of characteristic not $2$ and where $f(x) = (x - t_1) \ldots (x - t_ n)$ with $t_1, \ldots , t_ n \in k$ distinct and $n \geq 3$ an odd integer is not trivial. (Hint: Show that the ideal $(y, x - t_1)$ defines a nontrivial element of $\mathop{\mathrm{Pic}}\nolimits (A)$.)