Discriminant of a number field

The discriminant $D_K$ of a number field $K$ is the basis discriminant of any integral basis $(\omega_1,\ldots,\omega_n)$ :

D_K=\det\bigl(\operatorname{Tr}_{K/\mathbb Q}(\omega_i\omega_j)\bigr)_{i,j=1}^n.

It is a nonzero integer independent of the chosen integral basis. The absolute discriminant is $|D_K|$ .

Independence

An integral change of basis has determinant $\pm1$ . It changes the trace matrix to $P^{\mathsf T}TP$ , leaving its determinant unchanged. An arbitrary rational basis need not give the same number.

Imaginary quadratic case

For $K=\mathbb Q(\sqrt{-d})$ , $d>0$ square-free, the explicit integral basis gives

D_K=\begin{cases}-d,&d\equiv3\pmod4,\\-4d,&d\equiv1,2\pmod4.\end{cases}

Thus $D_{\mathbb Q(i)}=-4$ , while $D_{\mathbb Q(\sqrt{-3})}=-3$ .

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, §2, “Discriminants,” and Definition 2.32.