discriminant of a number field

《Number Fields》 · 第 18、24 页 · Chapter 2「Number Fields and Number Rings」，判别式定义及整数基推论

For embeddings $\sigma_1,\ldots,\sigma_n$ of a number field $K$, the discriminant of $\alpha_1,\ldots,\alpha_n$ is $\operatorname{disc}(\alpha_1,\ldots,\alpha_n)=|\sigma_i(\alpha_j)|^2$. The discriminant of an integral basis is an invariant of the ring $R$ and is denoted by $\operatorname{disc}(R)$; the text also writes $\operatorname{disc}(K)$ when $R$ is the ring of integers of $K$.