Showing that if $F$ an integral domain, $F[x_1,\dots,x_n]$ is an integral domain

Question

If $F$ is an integral domain, show that $F[x_1,\dots,x_n]$ is an integral domain.

Can anyone help out by giving me hints on how I should show that is an integral domain?

I would appreciate your help.

Answer 1

Hint: show this for $F[x]$ first (one indeterminate), considering the polynomial degree; then use the fact that $F[x_1,\dots,x_{n-1},x_n]=R[x_n]$ where $R=F[x_1,\dots,x_{n-1}]$