Proof that if an algebraic integer is rational, it is integer?

Question

There is a well-known fact that the intersection of and ℚ is ℤ. It is mentioned in many places, including Wikipedia, without proof. Does this theorem have a well-known name, and where can i find the proof?

Answer 1

In elementary contexts, one can refer to this as the monic case of the Rational Root test (RRT).

More generally, $\,\Bbb Z\,$ is integrally closed (in $\Bbb Q),\,$ since $ $ Euclidean $\,\Rightarrow$ PID $\Rightarrow$ gcd domains are integrally-closed (the proof of RRT in $\,\Bbb Z\,$ immediately generalizes to any domain with gcds).