We consider a non-zero totally real number $z$. We write $Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$ with $d$ minimal and $a_i \in \mathbb{Q}$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j )$ equals $t \left( z ^ { i + j } \right)$, and $P \in \mathrm{GL}_d(\mathbb{Q})$, $q_1, \ldots, q_d \in \mathbb{Q}$, $q_i > 0$ such that $S = P^T \cdot \operatorname{Diag}(q_1, \ldots, q_d) \cdot P$. We set $R = \operatorname{Diag}\left(\sqrt{q_1}, \ldots, \sqrt{q_d}\right) \cdot P$ and $M$ the companion matrix of $Z(X)$. Construct a symmetric matrix with rational coefficients for which $z$ is an eigenvalue.
We consider a non-zero totally real number $z$. We write $Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$ with $d$ minimal and $a_i \in \mathbb{Q}$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j )$ equals $t \left( z ^ { i + j } \right)$, and $P \in \mathrm{GL}_d(\mathbb{Q})$, $q_1, \ldots, q_d \in \mathbb{Q}$, $q_i > 0$ such that $S = P^T \cdot \operatorname{Diag}(q_1, \ldots, q_d) \cdot P$. We set $R = \operatorname{Diag}\left(\sqrt{q_1}, \ldots, \sqrt{q_d}\right) \cdot P$ and $M$ the companion matrix of $Z(X)$.
Construct a symmetric matrix with rational coefficients for which $z$ is an eigenvalue.