Factorization Process in a polynomial ring

Question

Reading the book "Field Theory" by S. Roman, in chapter $0$ I found the following problem:

Let $F$ be a field and consider the polynomial ring $F[x_1,x_2,\ldots]$ where $x_i^2 = x_{i-1}$. Show that the factorization process need not stop in this ring.

I am very confused with the question because I do not figure out what does the author means with "the process need not stop"?

Answer 1

You have posted incorrect statement(check your ring,again), he asked that factorisation process need not stop in ring $R=F[x_1,x_2,x_3,\ldots]$ such that $x_i^2 = x_{i-1}$.

So, now see that in the ring $R$,

$x_1=x_2.x_2=x_3.x_3.x_3.x_3=\underbrace{x_4.x_4\dots x_4}_{8\ times}=\dots$ and so on and it will never end. Also as all $x_i$ are irreducibles, so it is not a UFD and also factorisation never stops in it.