No quadratic field has discriminant 19

Show that there does not exist a quadratic field whose discriminant is 19.


Recall that the discriminant of \mathbb{Q}(\sqrt{D}) is 4D if D \not\equiv 1 mod 4 and is D if D \equiv 1 mod 4. In particular, the discriminant of a quadratic field is either 0 (if D \not\equiv 1 mod 4) or 1 (if D \equiv 1 mod 4) mod 4.

Since 19 \equiv 3 mod 4, it is not the discriminant of any quadratic field.

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