## Two quadratic fields with the same discriminant are equal

Let $E_1 = \mathbb{Q}(\sqrt{D_1})$ and $E_2 = \mathbb{Q}(\sqrt{D_2})$ be quadratic fields, with $D_1$ and $D_2$ squarefree integers. Show that if $E_1$ and $E_2$ have the same discriminant, then $E_1 = E_2$.

Let $\Delta_1$ and $\Delta_2$ be the discriminant of $E_1$ and $E_2$, respectively. We will summarize in the following table the possibilities for each $D_i$ mod 4 and the corresponding discriminants.

 $D_1 \mod 4$ $D_1 \mod 4$ $\Delta_1$ $\Delta_2$ 1 1 $D_1$ $D_2$ 1 2,3 $D_1$ $4D_2$ 2,3 1 $4D_1$ $D_2$ 2,3 2,3 $4D_1$ $4D_2$

We assert that the last two columns in each row are equal. Since $D_1$ and $D_2$ are squarefree, rows 2 and 3 are not possible, leaving only rows 1 and 4; in either case, $D_1 = D_2$. Hence $E_1 = E_2$.