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What is the statement of Noether normalization theorem you are referring to?

The one I know is the following, that you can find for instance in Shafarevich's book "Basic Algebraic Geometry I", page 65.

Theorem (Noether normalization). For an irreducible affine variety $X$ there exists a finite map $$\varphi \colon X \longrightarrow \mathbb{A}^n$$ to an affine space.

That means that any integral domain $A$ which is finitely generated over the field $k$ is integral over a subring isomorphic to a polynomial ring.

In particular, let $X \subset \mathbb{A}^{n+1}$ be any degree $d$ hypersurface. Then its "normal form" is given by a monic equation $$z_{n+1}^d + f_{d-1}z_{n+1}^{d-1}+ \ldots + f_1 z_{n+1}+f_0=0,$$ with $f_i \in k[z_1, \ldots, z_n]$, and the map $\varphi \colon X \longrightarrow \mathbb{A}^n$ in Noether theorem is simply the projection onto the first $n$ coordinates $z_1, \ldots, z_n$.

This shows how the two concepts of "normalization" are closely related.

What is the statement of Noether normalization theorem you are referring to?

The one I know is the following, that you can find for instance in Shafarevich's book "Basic Algebraic Geometry I", page 65.

Theorem (Noether normalization). For an irreducible, affine variety $X$ there exists a finite map $$\varphi \colon X \longrightarrow \mathbb{A}^n$$ to an affine space.

That means that any integral domain $A$ which is finitely generated over the field $k$ is integral over a subring isomorphic to a polynomial ring.

In particular, let $X \subset \mathbb{A}^{n+1}$ be any degree $d$ hypersurface. Then its "normal form" is given by a monic equation $$z_{n+1}^d + f_{d-1}z_{n+1}^{d-1}+ \ldots + f_1 z_{n+1}+f_0=0,$$ with $f_i \in k[z_1, \ldots, z_n]$, and the map $\varphi \colon X \longrightarrow \mathbb{A}^n$ in Noether theorem is simply the projection onto the first $n$ coordinates $z_1, \ldots, z_n$.

This shows how the two concepts of "normalization" are closely related.

What is the statement of Noether normalization theorem you are referring to?

The one I know is the following, that you can find for instance in Shafarevich's book "Basic Algebraic Geometry I", page 65.

Theorem (Noether normalization). For an irreducible affine variety $X$ there exists a finite map $$\varphi \colon X \longrightarrow \mathbb{A}^n$$ to an affine space.

That means that any integral domain $A$ which is finitely generated over the field $k$ is integral over a subring isomorphic to a polynomial ring.

So the two concepts of "normalization" are closely related.

If $X \subset \mathbb{A}^{n+1}$ is an hypersurface, its "normal form" is simply given by $$z_{n+1}^d + f_{n}z_{n+1}^{d-1}+ \ldots + f_1 z_{n+1}+f_0,$$ with $f_i \in k[z_2, \ldots, z_n]$, and the map $\varphi \colon X \longrightarrow \mathbb{A}^n$ in Noether theorem is the projection onto the first $n$ coordinates $z_1, \ldots, z_n$.

What is the statement of Noether normalization theorem you are referring to?

The one I know is the following, that you can find for instance in Shafarevich's book "Basic Algebraic Geometry I", page 65.

Theorem (Noether normalization). For an irreducible affine variety $X$ there exists a finite map $$\varphi \colon X \longrightarrow \mathbb{A}^n$$ to an affine space.

That means that any integral domain $A$ which is finitely generated over the field $k$ is integral over a subring isomorphic to a polynomial ring.

In particular, let $X \subset \mathbb{A}^{n+1}$ be any degree $d$ hypersurface. Then its "normal form" is given by a monic equation $$z_{n+1}^d + f_{d-1}z_{n+1}^{d-1}+ \ldots + f_1 z_{n+1}+f_0=0,$$ with $f_i \in k[z_1, \ldots, z_n]$, and the map $\varphi \colon X \longrightarrow \mathbb{A}^n$ in Noether theorem is simply the projection onto the first $n$ coordinates $z_1, \ldots, z_n$.

This shows how the two concepts of "normalization" are closely related.

What is the statement of Noether normalization theorem you are referring to? Tha one I know is the following, that you can find for instance in Shafarevich's book "Basic Algebraic Geometry I", page 65.

Theorem (Noether normalization). For an irreducible affine variety $X$ there exists a finite map $$\varphi \colon X \longrightarrow \mathbb{A}^n$$ to an affine space.

That means that any integral domain $A$ whici is finitely generated over the field $k$ is integral over a subring isomorphic to a polynomial ring.

So the two concepts of "normalization" are closely related.

What is the statement of Noether normalization theorem you are referring to?

The one I know is the following, that you can find for instance in Shafarevich's book "Basic Algebraic Geometry I", page 65.

Theorem (Noether normalization). For an irreducible affine variety $X$ there exists a finite map $$\varphi \colon X \longrightarrow \mathbb{A}^n$$ to an affine space.

That means that any integral domain $A$ which is finitely generated over the field $k$ is integral over a subring isomorphic to a polynomial ring.

So the two concepts of "normalization" are closely related.

If $X \subset \mathbb{A}^{n+1}$ is an hypersurface, its "normal form" is simply given by $$z_{n+1}^d + f_{n}z_{n+1}^{d-1}+ \ldots + f_1 z_{n+1}+f_0,$$ with $f_i \in k[z_2, \ldots, z_n]$, and the map $\varphi \colon X \longrightarrow \mathbb{A}^n$ in Noether theorem is the projection onto the first $n$ coordinates $z_1, \ldots, z_n$.