Exercise 4.4.9

Proof. $\text{∙}$ If $\alpha \in \mathbb{Z}$, $\alpha$ is a root of the monic polynomial $x-\alpha \in \mathbb{Z}[x]$, thus $\alpha$ is an algebraic integer.

$\text{∙}$ Conversely, let $\alpha \in \mathbb{Q}$ be an algebraic integer.

$\alpha$ is a root of $f=x^n+a_{n-1}{x}^{n-1}+\cdots +a_0$, where the coefficients $a_i$ are integers. Thus

that is

This implies $q\mid p^n$, where $q\wedge p=1$, thus $q\wedge p^n=1$. Hence $q\mid 1$, where $q>0$, thus $q=1$, and $\alpha =p∕q=p\in \mathbb{Z}$.

Conclusion: For all $\alpha \in \mathbb{Q}$, $\alpha$ is an algebraic integer iff $\alpha \in \mathbb{Z}$.

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