For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$. Let $\mathcal { T }$ be the set of $\alpha \in \mathcal { S }$ of degree 4. Show that $\mathcal { T }$ has a smallest element and calculate this number.
For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by
$$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$
We denote by $\mathcal { S }$ the set of real numbers $\alpha \in ] 1 , + \infty [$ which are also algebraic integers of degree at least 2 and which satisfy $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$. Let $\mathcal { T }$ be the set of $\alpha \in \mathcal { S }$ of degree 4. Show that $\mathcal { T }$ has a smallest element and calculate this number.