Relative fundamental discriminant?

Question

There is a well known bijection between the discriminant of quadratic extensions of $\mathbb Q$ and the fundamental discriminant of binary quadratic forms $ax^2 + bxy + cy^2, a,b,c\in \mathbb Q$.

Is there a similar relation between the relative discriminant of quadratic extensions of number fields, say $E/F$ over $\mathbb Q$, and discriminants of binary quadratic forms $ax^2 + bxy + cy^2, a,b,c\in F$?

Answer 1

Relative discriminants of quadratic extensions of a number field $F$ are ideals, which need not be principal. Discriminants of quadratic forms are elements of $F$ by definition.