# Computing local residues of traces of bilinear forms on algebraic number fields

This is a question about computing the local residues of a general symmetric bilinear form over $$\mathbb{Q}$$. I've been using Lam's Introduction to Quadratic Forms as a reference, but I'm stuck on the following.

Let $$\mathbb{Q}(\alpha)$$ be a degree $$d$$ extension of $$\mathbb{Q}$$. Given a non-zero element $$q=\sum_{i=0}^{d-1}b_i\alpha^i\in\mathbb{Q}(\alpha)$$, let $$\langle q\rangle\in\mathrm{W}(\mathbb{Q}(\alpha))$$ denote the Witt class of the symmetric, non-degenerate bilinear form $$\mathbb{Q}(\alpha)\times\mathbb{Q}(\alpha)\to\mathbb{Q}(\alpha)$$ given by $$(x,y)\mapsto qxy$$. Define $$\operatorname{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}\langle q\rangle\in\mathrm{W}(\mathbb{Q})$$ by post-composing $$\langle q\rangle$$ with the field trace $$\mathbb{Q}(\alpha)\to\mathbb{Q}$$. For each prime $$p$$ (including $$p=\infty$$), we have a local residue map $$\partial_p:\mathrm{W}(\mathbb{Q})\to\mathrm{W}(\mathbb{F}_p)$$ (or $$\mathrm{W}(\mathbb{R})$$ for $$p=\infty$$), and the weak Hasse-Minkowski principle says that the Witt class of $$\operatorname{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}\langle q\rangle$$ is determined the values $$\partial_p\operatorname{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}\langle q\rangle$$ for all $$p$$.

Question: Can I compute $$\partial_p\operatorname{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}\langle q\rangle$$ in terms of $$q$$ and the minimal polynomial of $$\alpha$$ over $$\mathbb{Q}$$? Or is a general formula too much to ask for?

For example, when $$q=1$$, this form is a trace form, and I believe a theorem of Taussky-Todd implies that $$\partial_\infty\operatorname{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}\langle 1\rangle\geq 0$$ (Corollary I.5.2, A survey of trace forms of algebraic number fields by Conner and Perlis).

Motivation: I'm writing up an (algebro-geometric) result, and I have a few equations involving terms of the form $$\partial_p\operatorname{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}\langle q\rangle$$. I can leave them this way, but I'm obviously not a number theorist and would thus like to record these values more explicitly if possible.