Finitely generated $k$-algebras beginner examples.

Question

I just found out about finitely generated $k$-algebras (where $k$ is a field). So it is an algebra $A$ for which we have a finite set of elements $(a_1,...,a_n)$ such that every element in $A$ can be expressed as $p(a_1,...,a_n)$ where $p$ is a polynomial $p \in k[x_1,...,x_n]$. This is pretty abstract for the moment so I am trying to illustrate with some examples. So I understand $k[x_1,...,x_n]$ itself is an example where the generators are $x_1,...,x_n$. It seems that in this particular case, the generators are even algebraically independent. What would be a nice example where the set of generators are not algebraically independent?

Answer 1

Consider $A = k[x,y]/(y-x^2)$. This is a finitely generated $k$-algebra where the generators, i.e. the images of $(x,y)$ in the quotient, are not algebraically independent. Can you see why not?