Non principal discriminant ideal

Question

Is there an extension of number fields $L/K$ such that the discriminant ideal $d_{L/K}$ isn't principal?

With discriminant I mean relative discriminant.

Answer 1

Yeah, this is fairly common.

Let $K=\mathbb Q(\sqrt[3]{-7})$ and $L=\mathbb Q(\sqrt{-3},\sqrt[3]{-7})$.

Then the relative discriminant of $L/K$ is the ideal $(3,1+\sqrt[3]{-7})$.

Proof by Sage:

K.<a> = NumberField(x^3+7)
t = K['t'].gen()
L.<b>=K.extension(t^2 +3)
L.relative_discriminant()