Irreducible elements in a factor ring

Question

Here is the problem 11.2.16 from Artin, Algebra:

$F$ is a field. Let $R$ be the ring $$\frac {F[u,v,y,x_1,x_2,x_3,...]}{(x_1y-uv,x{_2}{^2}-x_1,x{_3}{^2}-x_2,...)}.$$ Show that $u,v$ are irreducible elements in $R$.

1. The above problem can be diverted into the problem as follow:

$$R=\frac {F[u,v,y,x_1,...,x_n]}{(x_1y-uv,x{_2}{^2}-x_1,...,x{_n}{^2}-x_{n-1})}.$$ Show that $u$ is irreducible in $R$.

2. Since $$\frac {F[x_1,...,x_n]}{(x{_2}{^2}-x_1,...,x{_n}{^2}-x_{n-1})}=F[x]$$ by sending $x_i$ to $x^{2^{n-i}}$, $$R=\frac {F[u,v,y,x]}{(x^{2^{n-1}}y-uv)}.$$ Then I am stuck in the process to deal with the form as follow: $$u+p_1p_2=q(x^{2^{n-1}}y-uv)$$where $p_1,p_2,q\in F[u,v,x,y]$.

3. Maybe it is easier to firstly deal with $\dfrac {F[u,v,x,y]}{(xy-uv)}$. So is $\dfrac {F[u,v,x,y]}{(xy-uv)}$ isomorphic to a familiar ring we've know?

4. Maybe there exists another way to solve the problem. I expect your ideas. Thank you.