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.

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