Remark 25

Suppose that $A = K$ (maximal ordered field). Let L be an euclidean space of dimension $n$ over A, whose metric form is positive non-degenerate.

Let S be a non-degenerate affine quadric in L (§ 6, exerc. 25). If S admits a center $a$ and if one takes $a$ for origin in L, now prove that there exists an orthonormal basis $(e_i)$ for L such that, with respect to this basis, S has for equation $\lambda_1 \xi_1^2 + \cdots + \lambda_n \xi_n^2 = 1$. Moreover, if two orthonormal bases have this property, the elements $\lambda_i \in K$ corresponding to them are the same up to order.

If S has no center, show that there exists a point $b$ of S and (taking $b$ as origin) an orthonormal basis for L such that, with respect to this basis, S has the equation $\lambda_1 \xi_1^2 + \cdots + \lambda_{n-1} \xi_{n-1}^2 + \xi_n = 0$. (If $\overline{S}$ is the projective quadric such that $S = L \cap \overline{S}$, determine $b$ by the condition that the line passing through $b$ and $c$ be perpendicular to the tangent hyperplane to S at the point $b$, where c is the (at infinity) pole with respect to $\overline{S}$ of the hyperplane at infinity $H_0$).