# 数学代写|交换代数代写commutative algebra代考|MATH3303

## 数学代写|交换代数代写commutative algebra代考|The Elementary Case of Galois Theory

6.8 Definition and notation We will use the following notations when a group $G$ operates over a set $E$.

• For $x \in E, \operatorname{St}_G(x)=\operatorname{St}(x) \stackrel{\text { def }}{=}{\sigma \in G \mid \sigma(x)=x}$ designates the stabilizer of $x$.
• $G . x$ designates the orbit of $x$ under $G$, and we write $G . x=\left{x_1, \ldots, x_k\right}$ as an abbreviation for: $\left(x_1, \ldots, x_k\right)$ is an enumeration without repetition of $G \cdot x$, with $x_1=x$
• For $F \subseteq E, \operatorname{Stp}_G(F)$ or $\operatorname{Stp}(F)$ designates the pointwise stabilizer of $F$.
• If $H$ is a subgroup of $G$,
• we denote by $|G: H|$ the index of $H$ in $G$,
• we denote by $\operatorname{Fix}_E(H)=\operatorname{Fix}(H)=E^H$ the subset of elements fixed by $H$, ${x \in E \mid \forall \sigma \in H, \sigma(x)=x}$,
• writing $\sigma \in G / H$ means that we take an element $\sigma \in G$ in each left coset of $H$ in $G$.
When $G$ is a finite group operating over a ring $\mathbf{B}$, for $b \in \mathbf{B}$, we write
$$\operatorname{Tr}G(b)=\sum{\sigma \in G} \sigma(b), \mathrm{N}G(b)=\prod{\sigma \in G} \sigma(b), \text { and } \mathrm{C}G(b)(T)=\prod{\sigma \in G}(T-\sigma(b)) .$$
If $G . b=\left{b_1, \ldots, b_k\right}$, (the $b_i$ ‘s pairwise distinct), we write
$$\operatorname{Rv}{G, b}(T)=\prod{i=1}^k\left(T-b_i\right) .$$
This polynomial is called the resolvent of $b$ (relative to $G$ ). It is clear that $\left(\operatorname{Rv}_{G, b}\right)^r=$ $\mathrm{C}_G(b)$ with $r=\left|G: \mathrm{St}_G(b)\right|$.

Given an $\mathbf{A}$-algebra $\mathbf{B}$ we denote by $\operatorname{Aut}{\mathbf{A}}(\mathbf{B})$ the group of $\mathbf{A}$-automorphisms of $\mathbf{B}$. 6.9 Definition If $L$ is a strictly finite extension of $\mathbf{K}$, and a splitting field for a separable monic polynomial over $\mathbf{K}$, we say that $\mathbf{L}$ is a Galois extension of $\mathbf{K}$, we then denote $\operatorname{Aut}{\mathbf{K}}(\mathbf{L})$ by $\operatorname{Gal}(\mathbf{L} / \mathbf{K})$ and we say that it is the Galois group of the extension $\mathbf{L} / \mathbf{K}$.

Note well that in the definition of a Galois extension $\mathbf{L} / \mathbf{K}$, the fact that $\mathbf{L}$ is strictly finite (and not only finite) over $\mathbf{K}$ is implied.

## 数学代写|交换代数代写commutative algebra代考|Elimination Theory

Elimination theory concerns the systems of polynomial equations (or polynomial systems).

Such a system $\left(f_1, \ldots, f_s\right)$ in $\mathbf{k}\left[X_1, \ldots, X_n\right]=\mathbf{k}[X]$, where $\mathbf{k}$ is a discrete field, can admit some zeros in $\mathbf{k}^n$, or in $\mathbf{L}^n$, where $\mathbf{L}$ is an overfield of $\mathbf{k}$, or even an arbitrary

$\mathbf{k}$-algebra. The zeros depend only on the ideal $\mathfrak{a}=\left\langle f_1, \ldots, f_s\right\rangle$ of $\mathbf{k}[X]$ generated by the $f_i$ ‘s. We also call them the zeros of the ideal $a$.

Let $\pi: \mathbf{L}^n \rightarrow \mathbf{L}^r$ be the projection which forgets the last $n-r$ coordinates. If $V \subseteq \mathbf{L}^n$ is the set of zeros of $\mathfrak{a}$ on $\mathbf{L}$, we are interested in as precise a description as possible of the projection $W=\pi(V)$, if possible as zeros of a polynomial system in the variables $\left(X_1, \ldots, X_r\right)$.

Here intervenes in a natural way the elimination ideal (elimination of the variables $X_{r+1}, \ldots, X_n$ for the considered polynomial system), which is defined by $\mathfrak{b}=$ $\mathfrak{a} \cap \mathbf{k}\left[X_1, \ldots, X_r\right]$. Indeed every element of $W$ is clearly a zero of $\mathfrak{b}$.

The converse is not always true (and in any case not at all obvious), but it is true in some good cases: if $L$ is an algebraically closed field and if the ideal is in a Noether position (Théorểm 9.5).

A reassuring fact, and easy to establish by the considerations of linear algebra over discrete fields, is that the elimination ideal $\mathfrak{b}$ “does not depend on” the considered base field $\mathbf{k}$. More precisely, if $\mathbf{k}_1$ is an overfield of $\mathbf{k}$, we have the following results.

• The ideal $\left\langle f_1, \ldots, f_s\right\rangle_{\mathbf{k}_1}\left[X_1, \ldots, X_n\right]$ only depends on the ideal $a$ : it is the ideal $a_1$ of $\mathbf{k}_1\left[X_1, \ldots, X_n\right]$ generated by $a$.
• The ideal of elimination $\mathfrak{b}_1=\mathfrak{a}_1 \backslash \mathbf{k}_1\left[X_1, \ldots, X_r\right]$ only depends on $\mathfrak{b}$ : it is the ideal of $\mathbf{k}_1\left[X_1, \ldots, X_r\right]$ generated by $\mathfrak{b}$.
Elementary Elimination theory faces two obstacles.
The first is the difficulty of computing $\mathfrak{b}$ from $\mathfrak{a}$, i.e. of computing some finite generator set of $\mathfrak{b}$ from the polynomial system $\left(f_1, \ldots, f_s\right)$. This computation is rendered possible by the theory of the Gröbner bases, which we do not address in this work. In addition this computation is not uniform, unlike the computations linked to resultant theory.

## 数学代写|交换代数代写交换代数代考|伽罗瓦理论的初等情况

6.8定义和表示法当一个组$G$操作一个集合$E$时，我们将使用以下表示法

• 其中$x \in E, \operatorname{St}_G(x)=\operatorname{St}(x) \stackrel{\text { def }}{=}{\sigma \in G \mid \sigma(x)=x}$表示$x$的稳定器
• $G . x$表示$G$下$x$的轨道，我们将$G . x=\left{x_1, \ldots, x_k\right}$写为:$\left(x_1, \ldots, x_k\right)$是$G \cdot x$的一个不重复的枚举，其中$x_1=x$
• 对于$F \subseteq E, \operatorname{Stp}_G(F)$或$\operatorname{Stp}(F)$指定$F$的点向稳定器
• 如果$H$是$G$的一个子组，
• 用 $|G: H|$ 的指数 $H$ 在 $G$，
• 用 $\operatorname{Fix}_E(H)=\operatorname{Fix}(H)=E^H$ 由固定的元素的子集 $H$， ${x \in E \mid \forall \sigma \in H, \sigma(x)=x}$，
• writing $\sigma \in G / H$ 意思是我们取一个元素 $\sigma \in G$ 在每一个左胸 $H$ 在 $G$.
当 $G$ 有限群是否作用于环上 $\mathbf{B}$，为 $b \in \mathbf{B}$，我们写
$$\operatorname{Tr}G(b)=\sum{\sigma \in G} \sigma(b), \mathrm{N}G(b)=\prod{\sigma \in G} \sigma(b), \text { and } \mathrm{C}G(b)(T)=\prod{\sigma \in G}(T-\sigma(b)) .$$
如果 $G . b=\left{b_1, \ldots, b_k\right}$， ( $b_i$ 的成对不同)，我们写
$$\operatorname{Rv}{G, b}(T)=\prod{i=1}^k\left(T-b_i\right) .$$这个多项式叫做的解 $b$ (相对于 $G$ )。很明显 $\left(\operatorname{Rv}_{G, b}\right)^r=$ $\mathrm{C}_G(b)$ 用 $r=\left|G: \mathrm{St}_G(b)\right|$.

## 数学代写|交换代数代写对易代数代考|消去论

$\mathbf{k}$ -algebra。0只依赖于由$f_i$生成的$\mathbf{k}[X]$的理想$\mathfrak{a}=\left\langle f_1, \ldots, f_s\right\rangle$，我们也称它们为理想$a$的零

• 理想的$\left\langle f_1, \ldots, f_s\right\rangle_{\mathbf{k}_1}\left[X_1, \ldots, X_n\right]$只依赖于理想的$a$:它是$a$生成的$\mathbf{k}_1\left[X_1, \ldots, X_n\right]$的理想$a_1$
• 消除的理想$\mathfrak{b}_1=\mathfrak{a}_1 \backslash \mathbf{k}_1\left[X_1, \ldots, X_r\right]$只依赖于$\mathfrak{b}$:它是$\mathfrak{b}$生成的$\mathbf{k}_1\left[X_1, \ldots, X_r\right]$的理想
基本消除理论面临两个障碍。第一个是从$\mathfrak{a}$计算$\mathfrak{b}$的难度，即从多项式系统$\left(f_1, \ldots, f_s\right)$计算$\mathfrak{b}$的某个有限发电集的难度。这种计算是通过Gröbner基地的理论实现的，我们在本文中不讨论这个理论。此外，与与结果理论相关联的计算不同，这种计算不是均匀的。

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