Exercise 3.5.15* (Fundamental discriminants)

Error (sorry!)

Suppose that $d\equiv 0$ or $1(\mathit{mod}4)$ and that $d$ is not a perfect square. Then $d$ is called a fundamental discriminant (or reduced discriminant) if all binary quadratic forms of discriminant $d$ are primitive. Show that if $d\equiv 1(\mathit{mod}4)$ then $d$ is a fundamental discriminant if and only if $d$ is square-free. Show that if $d\equiv 0(\mathit{mod}4)$ then $d$ is a fundamental discriminant if and only if $d∕4$ is square-free and $d∕4\equiv 2$ or $3(\mathit{mod}4)$.