Let $f(X) \in \mathbb{Z}[X]$ be a monic polynomial. Suppose that $\alpha \in \mathbb{C}$ and $3\alpha$ are roots of $f$. (A) Show that $f(0) \neq 1$. (Hint: if $\zeta$ and $\zeta'$ are complex numbers satisfying monic polynomials in $\mathbb{Z}[X]$, then $\zeta\zeta'$ satisfies a monic polynomial in $\mathbb{Z}[X]$.) (B) Assume that $f$ is irreducible. Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of $f$. Let $\sigma$ be a field automorphism of $K$ such that $\sigma(\alpha) = 3\alpha$. Show that $\sigma$ has finite order and that $\alpha = 0$.
Let $f(X) \in \mathbb{Z}[X]$ be a monic polynomial. Suppose that $\alpha \in \mathbb{C}$ and $3\alpha$ are roots of $f$.\\
(A) Show that $f(0) \neq 1$. (Hint: if $\zeta$ and $\zeta'$ are complex numbers satisfying monic polynomials in $\mathbb{Z}[X]$, then $\zeta\zeta'$ satisfies a monic polynomial in $\mathbb{Z}[X]$.)\\
(B) Assume that $f$ is irreducible. Let $K$ be the smallest subfield of $\mathbb{C}$ containing all the roots of $f$. Let $\sigma$ be a field automorphism of $K$ such that $\sigma(\alpha) = 3\alpha$. Show that $\sigma$ has finite order and that $\alpha = 0$.