Is a polynomial ring over a UFD in countably many variables a UFD? Let $R$ be a UFD. It is well know that $R[x]$ is also a UFD, and so then is $R[x_1,x_2,\cdots,x_n]$ is a UFD for any finite number of variables. Is $R[x_1,x_2,\cdots,x_n,\cdots]$ in countably many variables also a UFD? If not, what about if we take $\tilde{R} = R[x_1,\cdots,x_n,\cdots]$ where each polynomial must be given in finitely many of the variables?

Hint: first show that for any $n\in\mathbb N$ and any $f\in R[x_1,x_2,\cdots,x_n]\subset R[x_1,x_2,\cdots]$, $f$ is irreducible in $R[x_1,x_2,\cdots]$ iff it's so in $R[x_1,x_2,\cdots,x_n]$; then notice any element of $R[x_1,x_2,\cdots]$ lies in $R[x_1,x_2,\cdots,x_n]$ for some $n$.
