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 set: $$M = \left( \begin{array} { c c c c c }
0 & 0 & \cdots & 0 & a _ { 0 } \\
1 & 0 & \ddots & 0 & a _ { 1 } \\
0 & \ddots & \ddots & \vdots & \vdots \\
\vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\
0 & \cdots & 0 & 1 & a _ { d - 1 }
\end{array} \right)$$ Compute the characteristic polynomial of $M$.
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 set:
$$M = \left( \begin{array} { c c c c c }
0 & 0 & \cdots & 0 & a _ { 0 } \\
1 & 0 & \ddots & 0 & a _ { 1 } \\
0 & \ddots & \ddots & \vdots & \vdots \\
\vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\
0 & \cdots & 0 & 1 & a _ { d - 1 }
\end{array} \right)$$
Compute the characteristic polynomial of $M$.