$\mathbb{Q}[x,1/x]$ is normal?

Question

Let $x$ be a transcendental.

I heard $\mathbb{Q}[x,1/x]$ is a normal domain. But I don't understand why.

Help me, thanks.

Answer 1

Hints:

1) Show that a UFD is a normal domain.

2) If $R$ is a normal domain, and $S \subseteq R$ is multiplicative, show that $S^{-1}R$ is a normal domain.

Alternatively, for this particular example, one can instead show that i) a PID is a normal domain, and ii) a localization of a PID is again a PID.