I try to understand the Noether normalization Lemma. But I'm totally lost. In our classes, we've noted for the lemma:
Let $\mathbb{K}$ be a field. $A$ be a finitely generated $\mathbb{K}$-algebra. Then there exists a polynomial $\mathbb{K}$-subalgebra $R:=\mathbb{K}[t_1,...,t_n] \subseteq A$ such that $A$ is integral over $R$.
Now, I know for example that the normalization from the quotient ring $\mathbb{K}[x,y]/(y^2-x^2-x^3)$ is $\mathbb{K}[t]$ via the parametrization $x=t^2-1$ and $y=t^3-t$, so that $ \mathbb{K}[t^2-1,t^3-t]\subset\mathbb{K}[t]$. Furthermore we get the ring $\mathbb{K}[t^2-1,t^3-t,t]=\mathbb{K}[x,y,\frac{y}{x}]$. But nevertheless I do not know which of these corresponds to the finitely generated $\mathbb{K}$-algebra $A$ and the subalgebra $R$ in my definition. I would say of course $A=\mathbb{K}[x,y]/(y^2-x^2-x^3)$ and $R=\mathbb{K}[t]$ but my intuition says it's wrong. How it would be right?
To my knowledge, these two uses of the word "normalization" are entirely unrelated. The polynomial subalgebra of $\mathbb{K}[x,y]/(y^2-x^2-x^3)$ is just $\mathbb{K}[x]$. Noether normalization says that we can view any finitely generated $\mathbb{K}$-algebra as an integral extension of a polynomial ring. In your example $y$ is integral over $\mathbb{K}[x]$.