Definition 1   Let $K$ be a number field, let $d_K$ be its discriminant and let $n=[K:\Rats]$ be the degree over $\Rats$ . The quantity: $$|\sqrt[n]{d_K}|$$ is called the root-discriminant of $K$ and it is usually denoted by $\operatorname{rd}_K$ .

Lemma 1   Let $E/F$ be an extension of number fields which is unramified at all finite primes. Then $\rd_E=\rd_F$ . In particular, the Hilbert class field of a number field has the same root-discriminant as the number field.

Proof. Notice that the relative discriminant ideal (or different) for $E/F$ is the ring of integers in $F$ . Therefore we have: $$|d_E|=|d_F|^{[E:F]}$$ The results follows by taking $[E:\Rats]$ -th roots on both sides of the previous equation.