I have got this question on my Commutative Algebra midterm exam.
Consider the $\mathbb{Q}$-subalgebra $C$ of $\mathbb{Q}[x,y]$ generated by two polynomials.
$f_1 = x^3+6x^2y+12xy^2+8y^3+2$
$f_2 = x^2+ 4xy+1$
Is $\mathbb{Q}[x,y]$ integral over $C$?
I have two potentially working ways to find an answer.
First: $x,y$ are integral,iff $C[x]$ and $C[y]$ are finitely generated $C$-modules.
Second Question from SE. Alas, I don't know what are the minimal prime ideals of $\mathbb{Q}[x,y]$
Let us notice that $f_1-2 = (x+2y)^3$ and $f_2-1 = (x+2y)^2 - y^2$.
Why does this mean that $\mathbb{Q}[x,y]$ is integral over $С$?