Remark 3

Suppose that $A$ is a maximal ordered field. Then the ring $\mathfrak{B}$ is isomorphic to $\mathbf{Z}$ (prop. 5), the integer corresponding to the type of a form of signature $(s,\ t)$ being $s - t$ (ibid.). Since the tensor product of two forms $Q$, $Q'$ of signatures $(s,\ t)$, $(s',\ t')$ is a form of dimension $(s + t)\,(s' + t')$, it follows, by means of an elementary calculus, that the signature of $Q \otimes Q'$ is $(ss' + tt',\ st' + ts')$.