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# 数学代写|交换代数代写Commutative Algebra代考|Hilbert’s Nullstellensatz

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## 数学代写|交换代数代写Commutative Algebra代考|Hilbert’s Nullstellensatz

In this section we illustrate the importance of the resultant by showing how Hilbert’s Nullstellensatz can be deducted from it. We will use a generalization of the basic elimination Lemma 7.5.
The Algebraic Closure of $\mathbb{Q}$ and of Finite Fields
Let $\mathbf{K} \subseteq \mathbf{L}$ be discrete fields. We say that $\mathbf{L}$ is an algebraic closure of $\mathbf{K}$ if $\mathbf{L}$ is algebraic over $\mathbf{K}$ and algebraically closed.

The reader will concede that $\mathbb{Q}$ and the fields $\mathbb{F}_p$ possess an algebraic closure. This will be discussed in further detail in Sect. VI-1, especially with Theorem VI-1.18.

The Classical Nullstellensatz (Algebraically Closed Case)
The Nullstellensatz is a theorem which concerns the systems of polynomial equations over a discrete field. Very informally, its meaning can be described as follows: a geometric statement necessarily possesses an algebraic certificate. Or even: a proof in commutative algebra can (almost) always be summarized by simple algebraic identities if it is sufficiently general.

If we have discrete fields $\mathbf{K} \subseteq \mathbf{L}$, and if $(\underline{f})=\left(f_1, \ldots, f_s\right)$ is a system of polynomials in $\mathbf{K}\left[X_1, \ldots, X_n\right]=\mathbf{K}[\underline{X}]$, we say that $\left(\xi_1, \ldots, \xi_n\right)=(\underline{\xi})$ is a zero of $(\underline{f})$ in $\mathbf{L}^n$, or a zero of $(\underline{f})$ with coordinates in $\mathbf{L}$, if the equations $f_i(\underline{\xi})=0$ are satisfied. Let $\mathfrak{f}=\left\langle f_1, \ldots, \overline{f_s}\right\rangle_{\mathbf{K}}[\underline{X}]$. Then, all the polynomials $g \in \mathfrak{f}$ are annihilated in such a $(\underline{\xi})$. We therefore equally refer to $(\underline{\xi})$ as a zero of the ideal $f$ in $\mathbf{L}^n$ or as having coordinates in $\mathbf{L}$.
We begin with an almost obvious fact.
9.1 Fact Let $\mathbf{k}$ be a commutative ring and $h \in \mathbf{k}[X]$ a monic polynomial of degree $\geqslant 1$.

• If some multiple of $h$ is in $\mathbf{k}$, this multiple is null.
• Let $f$ and $g \in \mathbf{k}[X]$ of respective formal degrees $p$ and $q$. If $h$ divides $f$ and $g$, then $\operatorname{Res}_X(f, p, g, q)=0$.

## 数学代写|交换代数代写Commutative Algebra代考|The Formal Nullstellensatz

We now move onto a formal Nullstellensatz, formal in the sense that it applies (in classical mathematics) to an arbitrary ideal over an arbitrary ring. Nevertheless to have a constructive statement we will be content with a polynomial ring $\mathbb{Z}[\underline{X}]$ for our arbitrary ring and a finitely generated ideal for our arbitrary ideal.

Although this may seem very restrictive, practice shows that this is not the case because we can (almost) always apply the method of undetermined coefficients to a commutative algebra problem; a method which reduces the problem to a polynomial problem over $\mathbb{Z}$. An illustration of this will be given next.

Note that to read the statement, when we speak of a zero of some $f_i \in \mathbb{Z}[\underline{X}]$ over a ring $\mathbf{A}$, one must first consider $f_i$ modulo $\operatorname{Ker} \varphi$, where $\varphi$ is the unique homomorphism $\mathbb{Z} \rightarrow \mathbf{A}$, with $\mathbf{A}_1 \simeq \mathbb{Z} / \operatorname{Ker} \varphi$ as its image. This thus reduces to a polynomial $\overline{f_i}$ of $\mathbf{A}_1[\underline{X}] \subseteq \mathbf{A}[\underline{X}]$.
9.9 Theorem (Nullstellensatz over $\mathbb{Z}$, formal Nullstellensatz) Let $\mathbb{Z}[\underline{X}]=$ $\mathbb{Z}\left[X_1, \ldots, X_n\right]$. Consider $g, f_1, \ldots, f_s$ in $\mathbb{Z}[\underline{X}]$

1. For the system $\left(f_1, \ldots, f_s\right)$ the following properties are equivalent.
a. $1 \in\left\langle f_1, \ldots, f_s\right\rangle$.
b. The system does not admit a zero on any nontrivial discrete field.
c. The system does not admit a zero on any finite field or on any finite extension of $\mathbb{Q}$.
d. The system does not admit a zero on any finite field.
2. The following properties are equivalent.
a. $\exists N \in \mathbb{N}, g^N \in\left\langle f_1, \ldots, f_s\right\rangle$.
b. The polynomial $g$ is annihilated at the zeros of the system $\left(f_1, \ldots, f_s\right)$ on any discrete field.
c. The polynomial $g$ is annihilated at the zeros of the system $\left(f_1, \ldots, f_s\right)$ on every finite field and on every finite extension of $\mathbb{Q}$.
d. The polynomial $g$ is annihilated at the zeros of the system $\left(f_1, \ldots, f_s\right)$ on every finite field.

## 数学代写|交换代数代写Commutative Algebra代考|Hilbert’s Nullstellensatz

Nullstellensatz是一个关于离散域上多项式方程组的定理。非常非正式地，它的含义可以描述如下:一个几何陈述必须拥有一个代数证明。或者甚至:交换代数中的证明(几乎)总是可以用简单的代数恒等式来概括，如果它足够普遍的话。

9.1事实设$\mathbf{k}$为交换环，$h \in \mathbf{k}[X]$为次为$\geqslant 1$的一元多项式。

## 数学代写|交换代数代写Commutative Algebra代考|The Formal Nullstellensatz

9.9定理(Nullstellensatz over $\mathbb{Z}$，正式的Nullstellensatz)设$\mathbb{Z}[\underline{X}]=$$\mathbb{Z}\left[X_1, \ldots, X_n\right]。考虑g, f_1, \ldots, f_s$$\mathbb{Z}[\underline{X}]$

A. $1 \in\left\langle f_1, \ldots, f_s\right\rangle$;
b.系统在任何非平凡离散域上都不允许存在零。
c.系统不允许在任何有限域或$\mathbb{Q}$的任何有限扩展上存在零。
d.系统在任何有限域上都不允许存在零。

A. $\exists N \in \mathbb{N}, g^N \in\left\langle f_1, \ldots, f_s\right\rangle$;
b.多项式$g$在任意离散场上的系统$\left(f_1, \ldots, f_s\right)$的零点处被湮灭。
c.多项式$g$在系统$\left(f_1, \ldots, f_s\right)$的每一个有限域和$\mathbb{Q}$的每一个有限扩展的零点处被湮灭。
d.多项式$g$在每个有限域的系统$\left(f_1, \ldots, f_s\right)$的零点处被湮灭。

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