Lorenzini's “Invitation to arithmetic geometry”, 2nd exercise

Question

I have some trouble trying to prove that

\begin{equation} \mathbb{Z}\left[\frac{2+i}{5}\right]\cap \mathbb{Q}=\mathbb{Z} \end{equation}

which is the second exercise of Dino Lorenzini's "An invitation to arithmetic geometry".

Thanks for your help.

Answer 1

First of all I'm apologized for my bad English.

Called $x=\frac{2+i}{5}$ and defines $\mathbb{Z}[x]=\{a+bx \vert a,b \in \mathbb{Z} \}$.

Observe what $\mathbb{Z} \subseteq \mathbb{Z}[x] $ with $ b=0$ then

$\mathbb{Z} \subseteq \mathbb{Z}[x] \cap \mathbb{Q} $.

Now take $ z \in \mathbb{Z}[x] \cap \mathbb{Q} $ then $ z=a+bx$ with $a,b \in \mathbb{Z}$ and $ z \in \mathbb{Q}$ so the only way to belong to both sets is that $ z=a $.

therefore it has to be $ \mathbb{Z} = \mathbb{Z}[x] \cap \mathbb{Q} $.