Let and be quadratic fields, with and squarefree integers. Show that if and have the same discriminant, then .
Let and be the discriminant of and , respectively. We will summarize in the following table the possibilities for each mod 4 and the corresponding discriminants.
We assert that the last two columns in each row are equal. Since and are squarefree, rows 2 and 3 are not possible, leaving only rows 1 and 4; in either case, . Hence .