Remark 6

Let $K$ be a maximal ordered field, $L$ a simple algebra of finite rank over $K$. If $(s, t)$ is the signature of the symmetric bilinear form $(x, y) \to \operatorname{Tr}_{L/K}(xy)$ on $L$, now prove that one has: $s - t = m$ if $L$ is isomorphic to a matrix algebra of order $m$ over $K$; $s - t = 0$ if $L$ is isomorphic to a matrix algebra over the field $K(i)$; $s - t = -2m$ if $L$ is isomorphic to a matrix algebra of order $m$ over the quaternion field over $K$.