Concern about algebraic integer

Question

To be honest, I don't figure out how to attack this problem:

Let $\alpha$ an algebraic integer i.e. there is a monic polynomial $f(x) \in \mathbb{Z}[x]$ s.t. $f(\alpha)=0$.

Let $R:=\mathbb{Z}[\alpha]$. For some positive integer $m$, prove that $R/mR$ is finite and determine its order.

I'll appreciate any help/hint.

Thanks.

Answer 1

Sorry for my late reply.

Following your idea: $F[\alpha]=\{a_0+a_1\alpha+\cdots+a_{n-1}\alpha^{n-1}\mid a_j \in \mathbb{Z}\}$. It looks like $\{\alpha^0,\alpha, \alpha^2,...,\alpha^{n-1}\}$ is a basis for $F[\alpha]$. Moreover, $F[\alpha]\cong \mathbb{Z}^n$ under this basis. Thus, given a positive integer $m$, $$ R/mR \cong \mathbb{Z}^n/m\mathbb{Z}^n\cong \frac{\mathbb{Z} \oplus \mathbb{Z}\oplus \cdots \oplus \mathbb{Z}}{m \mathbb{Z}\oplus m\mathbb{Z}\oplus\cdots \oplus m \mathbb{Z}} \quad \textrm{ By invariant factor form}. $$ Therefore, $$ \left|R/mR\right|=m^n. $$ I would like to be more precisely in each step, it's like the sketch of a possible proof. Thanks so much for any extra information that you see might be needed.