SCHWEITZER 2017 Problem 2

Prove that a field $K$ can be ordered if and only if every $A\in M_n(K)$ symmetric matrix can be diagonalized over the algebraic closure of $K$ . (In other words, for all $n\in\mathbb{N}$ and all $A\in M_n(K)$ , there exists an $S\in GL_n(\overline{K})$ for which $S^{-1}AS$ is diagonal.)