Show $f(x,y)=x^7+yx^5+yx^3+3yx+y\in R[x,y]$ is irreducible in $R[x,y]$.

Question

Let $R$ be a UFD and consider $$ f(x,y) = x^7 + yx^5 + yx^3 + 3yx + y \in R[x,y]. $$ Show that $f(x,y)$ is irreducible in $R[x,y]$.

Proof: Note $R[x,y]=R[y][x]$. Let $S=R[y]$. We will show $f(x,y)$ is irreducible in $S[x]$. By Eisenstein Criterion, let $p=y$. (How do I know y is prime in $S[x]$?) We can see

• $y | a_0=a_1=a_2=a_3=y, a_4=0$
• $y\not | a_5=1$
• $y^2 \not | a_0=y$ Hence $f(x,y)$ is irreducible in $S[x]$.

Am I done?