definition of discriminant and traces of number field. Let $K=\Bbb Q [x]$ be a number field, $A$ be the ring of integers of $K$.
Let $(x_1,\cdots,x_n)\in A^n$. In usual, what does it mean $D(x_1,\cdots,x_n)$? Either $\det(Tr_{\Bbb K/ \Bbb Q} (x_ix_j))$ or $\det(Tr_{A/ \Bbb Z} (x_ix_j))$? Or does it always same value? I searched some definitions, but it is not explicitly stated.

I'm not entirely certain what $\operatorname{Tr}_{A/\mathbb{Z}}$ is, but the notation $D(x_1,\dots,x_n)$ or $\Delta(x_1,\dots,x_n)$ usually means the discriminant of $K$ with respect to the basis $x_1,\dots,x_n$, so I would say it's most likely the former.
After all, $x_1,\dots,x_n\in A\subset K$, so it still makes sense to talk about the trace of these as elements of $K$, and that is what the definition of the discriminant of $K$ with respect to a basis is (assuming that $x_1,\dots,x_n$ are indeed a basis for $K/\mathbb{Q}$!).
