The anisotropic part of a quadratic form over a number field (Q2673997)

Let \(q\) be a non-degenerate quadratic form with coefficients in a number field \(K\). If there exists a non-zero vector \(v\) such that \(q(v)=0\), then the form is said to be isotropic. Witt's decomposition theorem states that any quadratic form over a field is isometric to \(q_a\perp w\times \langle 1,-1\rangle\), where \(q_a\) is the anisotropic part of \(q\) and \(w \geq 0\) is the Witt index of \(q\). This paper outlines an explicit technique for computing the anisotropic part of a given quadratic form over an arbitrary number field \(K\). Specifically, for a given quadratic form \(q\), the algorithm returns an element \(\alpha\in K\) such that the anisotropic dimension of \(q\) is greater than that of \(q \perp \langle -\alpha \rangle\) (as detailed in Section 5). Such an element \(\alpha\) must have a prescribed signature with respect to different orderings in \(K\), and the method for finding such an element is contained in Section 4. Finally, the paper concludes with a concrete example over \(K = \mathbb{Q}(\sqrt{-7})\).