Proposition 5

*We suppose that A is a maximal ordered field. Let Q be a non-degenerate quadratic form of finite dimension over A, and $(s, t)$ its signature ($§ 7$, no 2, def. 2). Then the type of Q is $(s - t).T_1$, and the group $\mathfrak{W}$ of types of quadratic forms over A is an infinite monogenic group generated by $T_1$.

Indeed, since $A^*/(A^*)^2$ is of order 2 and $(-1).T_1 = -T_1$, $\mathfrak{W}$ is generated by $T_1$ and is therefore monogenic. For every $n > 0$, $n.T_1$ is the type of the positive non-degenerate quadratic forms of dimension $n$; since these forms are not neutral, we have $n.T_1 \neq 0$, which shows that $\mathfrak{W}$ is infinite. Finally, a form of signature $(s, t)$ is isomorphic, with the notation of prop. 4, to the direct sum of $s$ forms $Q_1$ and of $t$ forms $- Q_1$ (§ 7, n° 2, th. 1); it follows that its type is $(s - t). T_1$.