Posted on Categories:Commutative Algebra, 交换代数, 数学代写

# 数学代写|交换代数代写Commutative Algebra代考|MA8202 Norms and Traces

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|交换代数代写Commutative Algebra代考|Norms and Traces

Let $A$ be a ring and let $B$ be an $A$-algebra; let us assume that $B$ is a free finitely generated A-module. An important case of this situation arises when A is a field and $B$ is a finite extension of $A$.

Since $A$ is commutative, all bases of $B$ have the same cardinality, equal to $n=\operatorname{rk}{\mathrm{A}}(\mathrm{B})$, by definition of the rank of $\mathrm{B}$. For any $b \in \mathrm{B}$, multiplication by $b$ gives rise to an endomorphism $x \mapsto b x$ of the A-module B; one defines the characteristic polynomial of $b$ as the characteristic polynomial $\mathrm{P}_b$ of this endomorphism. This polynomial is defined either as $\operatorname{det}\left(X-M_b\right)$, where $M_b \in \operatorname{Mat}_n(\mathrm{~A})$ is the matrix of the multiplication by $b$, written in any basis of B. Let us write $$\mathrm{P}_b(\mathrm{X})=\mathrm{X}^n-\operatorname{Tr}{\mathrm{B} / \mathrm{A}}(b) \mathrm{X}^{n-1}+\cdots+(-1)^n \mathrm{~N}{\mathrm{B} / \mathrm{A}}(b) .$$ Definition (4.7.1). – The elements $\operatorname{Tr}{\mathrm{B} / \mathrm{A}}(b)$ and $\mathrm{N}_{\mathrm{B} / \mathrm{A}}(b)$ are called the trace and the norm of $b$ with respect to $\mathrm{A}$.

Note that $\operatorname{Tr}{\mathrm{B} / \mathrm{A}}(b)$ and $\mathrm{N}{\mathrm{B} / \mathrm{A}}(b)$ are the trace and the norm of the matrix $U_b \in \operatorname{Mat}_n(\mathrm{~A})$ of multiplication by $b$, written in any basis of B.

Remark (4.7.2). – Using notions that we will introduce later, an intrinsic definition is possible: consider the $\mathrm{A}[\mathrm{X}]$-module $\mathrm{B}[\mathrm{X}]=\mathrm{A}[\mathrm{X}] \otimes_{\mathrm{A}} \mathrm{B}$; it is free of rank $n$ and multiplication by $X-b$ in $B[X]$ induces an endomorphism of the free rank-1 module $\Lambda^n B[X]$, which is precisely multiplication by $P_b(X)$. Moreover, $\Lambda^n(\mathrm{~B})$ is a free A-module of rank 1 and multiplication by $b$ in B induces multiplication by $\mathrm{N}_{\mathrm{B} / \mathrm{A}}(b)$ on $\Lambda^n(\mathrm{~B})$.

## 数学代写|交换代数代写Commutative Algebra代考|Transcendence Degree

Proposition (4.8.1). – Let $\mathrm{K} \rightarrow \mathrm{L}$ be a field extension and let $\left(x_i\right){i \in \mathrm{I}}$ be a family of elements of $\mathrm{L}$. The following propositions are equivalent: (i) For every non-zero polynomial $\mathrm{P} \in \mathrm{K}\left[\left(X_i\right){i \in \mathrm{I}}\right]$, one has $\mathrm{P}\left(\left(x_i\right)\right) \neq 0$;
(ii) The canonical morphism of $\mathrm{K}$-algebras from $\mathrm{K}\left[\left(\mathrm{X}_i\right)_i\right]$ to $\mathrm{L}$ such that $\mathrm{X}_i \mapsto x_i$ is injective;
(iii) There exists a $\mathrm{K}$-morphism of fields $\mathrm{K}\left(\left(\mathrm{X}_i\right)_i\right) \rightarrow \mathrm{L}$ such that $\mathrm{X}_i \mapsto x_i$.
Proof. – Let $\varphi: \mathrm{K}\left[\left(X_i\right)_i\right] \rightarrow \mathrm{L}$ be the unique morphism of K-algebras such that $\varphi\left(X_i\right)=x_i$ for all $i \in \mathrm{I}$.

Condition (i) means that $\varphi(\mathrm{P}) \neq 0$ if $\mathrm{P} \neq 0$; it is thus equivalent to the equality $\operatorname{Ker}(\varphi)=0$ of (ii).

Assume (ii). Note that $\mathrm{K}\left[\left(\mathrm{X}i\right){i \in \mathrm{I}}\right]$ is an integral domain, and its field of fractions is $\mathrm{K}\left(\left(X_i\right)\right]$. If $\varphi$ is injective, then, since $L$ is a field, it extends to a morphism of fields from $\mathrm{K}\left[\left(\mathrm{X}_i\right)\right]$ to $\mathrm{L}$, hence (iii).

Finally, assume that (iii). Let $\psi: \mathrm{K}\left(\left(\mathrm{X}_i\right)\right) \rightarrow \mathrm{L}$ be a $\mathrm{K}$-morphism of fields such that $\psi\left(X_i\right)=x_i$ for all $i$. The restriction of $\psi$ to $\mathrm{K}\left[\left(\mathrm{X}_i\right)\right]$ is the morphism of K-algebras which was denoted by $\varphi$. Since $\psi$ is injective, $\varphi$ is injective too, hence (ii).

## 数学代写|交换代数代写Commutative Algebra代考|Transcendence Degree

(ii) 的规范态射 $\mathrm{K}$-代数来自 $\mathrm{K}\left[\left(\mathrm{X}_i\right)_i\right]$ 臸这样 $\mathrm{X}_i \mapsto x_i$ 是单射的;
(iii) 存在一个K-域的态射 $\mathrm{K}\left(\left(\mathrm{X}_i\right)_i\right) \rightarrow \mathrm{L}$ 这样 $\mathrm{X}_i \mapsto x_i$.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。