# Finitely generated $k$-algebras beginner examples.

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?

• Any quotient of your example; a simple example would be $k[x,y]/(xy)$. Commented Jan 27, 2019 at 0:16
• Maybe this is to concrete for you, but how about $\mathbb{Q}[\sqrt{2}, \pi]$? The generators $\sqrt{2}, \pi$ are not algebraically independent, because $$(\sqrt{2})^2 + 0 \cdot \pi - 2 =0.$$ Commented Jan 27, 2019 at 0:20
• @SeverinSchraven The $\pi$ in that example is redundant, why include it?
– jgon
Commented Apr 19, 2019 at 0:17
• @jgon Because I like it. Also because I think it is better from a pedagocial point of view to to include two elements when talking about algebraic dependence. Feel free to post an example which is more to your liking. Commented Apr 19, 2019 at 1:34

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?
• But the quotient is basically just $k[x]$. Better perhaps to use $y^3-x^2$ so it isn't trivial. Commented Jan 27, 2019 at 1:19
• @Mr.chip Yes, thank you. The generators are $\bar{x}$ and $\bar{y}$ but there is an algebraic relation $\bar{y}=\bar{x}^2$ so they are not algebraically independent. So basically every finitely generated $k$-Algebra is isomorphic to a quotient of $k[x_1,...,x_n]$ if $A$ has generator $a_1,...,a_n$? Commented Jan 27, 2019 at 12:23