These days I much prefer a different proof of this fact, attributed to Conway: https://www.youtube.com/watch?v=wNOtOPjaLZs -- I'm actually about half-way through teaching it to my kids right now!
In commutative algebra one says "the ring Z is integrally closed in its field of fractions Q", or short "Z is normal".[1]
[1] https://en.wikipedia.org/wiki/Integrally_closed_domain#Norma...