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 $\alpha$ be an element of $\mathcal { S }$ and let $\gamma \in C ( \alpha )$ have modulus 1. (a) Show that the minimal polynomial of $\alpha$ is reciprocal and that $\frac { 1 } { \alpha }$ is a conjugate of $\alpha$. (b) Show that $\gamma$ is not a root of unity. (c) Show that all conjugates of $\alpha$ other than $\frac { 1 } { \alpha }$ have modulus 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 $\alpha$ be an element of $\mathcal { S }$ and let $\gamma \in C ( \alpha )$ have modulus 1.\\
(a) Show that the minimal polynomial of $\alpha$ is reciprocal and that $\frac { 1 } { \alpha }$ is a conjugate of $\alpha$.\\
(b) Show that $\gamma$ is not a root of unity.\\
(c) Show that all conjugates of $\alpha$ other than $\frac { 1 } { \alpha }$ have modulus 1.