## 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.