I would like to prove or disprove that $S = \mathbb{C}[x, y]/(y^4 + y^3 + y^2 + y + 1)$ is integral over $\mathbb{C}[x]$.
Just using the definition of integrality might not be the correct approach, because then I have to find a monic polynomial $f \in \mathbb{C}[x]$ such that any element in $\mathbb{C}[x, y]/(y^4 + y^3 + y^2 + y + 1)$, which is a polynomial of $2$ variables, is a root of $f$. My idea is to show that $\mathbb{C}[x, y]/(y^4 + y^3 + y^2 + y + 1)$ is (or not) finitely generated as a $\mathbb{C}[x]$-module. If this is not true, is $S$ at least integral over $\mathbb{C}$ ? Unfortunately, I don't know how to proceed. I am grateful for any help.