grandes-ecoles 2019 Q18

grandes-ecoles · France · x-ens-maths1__mp Number Theory Algebraic Number Theory and Minimal Polynomials
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 $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.
Show that the degree of every element of $\mathcal { S }$ is an even integer, greater than or equal to 4.
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 $\max _ { \gamma \in C ( \alpha ) } | \gamma | = 1$.

Show that the degree of every element of $\mathcal { S }$ is an even integer, greater than or equal to 4.