We say that a complex number $z$ is totally real (resp. totally positive) if there exists a non-zero polynomial $P ( X )$ with rational coefficients such that:
(i) $z$ is a root of $P$, and
(ii) all roots of $P$ are in $\mathbb { R }$ (resp. in $\mathbb { R } _ { + }$).
Let $x$ be a complex number. Show that $x$ is totally real if and only if $x ^ { 2 }$ is totally positive.