TITLE: Showing that $\overline{\mathbb{Z}}/n\overline{\mathbb{Z}}$ is infinite
QUESTION [7 upvotes]: Let $\overline{\mathbb{Z}}$ be the ring of algebraic integers. Since $n\overline{\mathbb{Z}}$ is an ideal of $\overline{\mathbb{Z}}$, we can form the quotient ring $\overline{\mathbb{Z}}/n\overline{\mathbb{Z}}$. I want to prove that it is infinite.
The exercise has a hint to consider $x_i=n^{1/{2^i}}$, for all $i\geq 1$. However it is not clear to me why would someone think about these numbers nor why they are distinct in the quotient ring.

REPLY [2 votes]: $a=b$ in $\overline{\mathbb{Z}}/n\overline{\mathbb{Z}}$ iff $\frac{a-b}{n}$ is an algebraic integer, 
Let $c=\frac{n^{2^{-i}}-n^{2^{-j}}}{n} $ with $j  >i$
since $n^{1-2^{-i}}$ is an algebraic integer, if $c$ was an algebraic integer then $cn^{1-2^{-i}}=1-n^{2^{-j}-2^{-i}}$ thus $n^{2^{-j}-2^{-i}}$ and $(n^{2^{-j}-2^{-i}})^{2^j} = n^{1-2^{j-i}}$ would be algebraic integers, a contradiction because it is immediate a non-integer rational number cannot be a root of a monic integer polynomial.