root-discriminant

Definition 1.

Let $K$ be a number field, let $d_K$ be its discriminant and let $n\mathrm{=}\mathrm{[}K\mathrm{:}\mathrm{Q}\mathrm{]}$ be the degree over $\mathrm{Q}$. The quantity:

$$|\sqrt[n]{d_K}|$$

is called the root-discriminant of $K$ and it is usually denoted by ${\mathrm{rd}}_K$.

The following lemma is one of the motivations for the previous definition:

Lemma 1.

Let $E\mathrm{/}F$ be an extension of number fields which is unramified at all finite primes. Then ${\mathrm{rd}}_E\mathrm{=}{\mathrm{rd}}_F$. 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 $[E:\mathbb{Q}]$-th roots on both sides of the previous equation.∎