# 线性代数网课代修|交换代数代写Commutative Algebra代考|MAT4200

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|Unique Factorization and Resultants

This definition says that if a nonconstant polynomial $f$ is irreducible over $k$, then up to a constant multiple, its only nonconstant factor is $f$ itself. Also note that the concept of irreducibility depends on the field. For example, $x^{2}+1$ is irreducible over $\mathbb{Q}$ and $\mathbb{R}$, but, over $\mathbb{C}$, we have $x^{2}+1=(x-i)(x+i)$. Every polynomial is a product of irreducible polynomials as follows. Proposition 2. Every nonconstant polynomial $f \in k\left[x_{1}, \ldots, x_{n}\right]$ can be written as a product of polynomials which are irreducible over $k$.

Proof. If $f$ is irreducible over $k$, then we are done. Otherwise, we can write $f=g h$, where $g, h \in k\left[x_{1}, \ldots, x_{n}\right]$ are nonconstant. Note that the total degrees of $g$ and $h$ are less than the total degree of $f$. Now apply this process to $g$ and $h$ : if either fails to be irreducible over $k$, we factor it into nonconstant factors. Since the total degree drops each time we factor, this process can be repeated at most finitely many times. Thus, $f$ must be a product of irreducibles.

In Theorem 5 we will show that the factorization of Proposition 2 is essentially unique. But first, we have to prove the following crucial property of irreducible polynomials.

Theorem 3. Let $f \in k\left[x_{1}, \ldots, x_{n}\right]$ be irreducible over $k$ and suppose that $f$ divides the product $g h$, where $g, h \in k\left[x_{1}, \ldots, x_{n}\right]$. Then $f$ divides $g$ or $h$.

Proof. We will use induction on the number of variables. When $n=1$, we can use the GCD theory developed in $\S 5$ of Chapter 1. If $f$ divides $g h$, then consider $p=$ $\operatorname{GCD}(f, g)$. If $p$ is nonconstant, then $f$ must be a constant multiple of $p$ since $f$ is irreducible, and it follows that $f$ divides $g$. On the other hand, if $p$ is constant, we can assume $p=1$, and then we can find $A, B \in k\left[x_{1}\right]$ such that $A f+B g=1$ (see Proposition 6 of Chapter $1, \S 5$ ). If we multiply this by $h$, we get $$h=h(A f+B g)=A h f+B g h .$$ Since $f$ divides $g h, f$ is a factor of $A h f+B g h$, and, thus, $f$ divides $h$. This proves the case $n=1$.

## 线性代数作业代写linear algebra代考|Resultants and the Extension Theorem

In this section we will prove the Extension Theorem using the results of $\S$. Our first task will be to adapt the theory of resultants to the case of polynomials in $n$ variables. Thus, suppose we are given $f, g \in k\left[x_{1}, \ldots, x_{n}\right]$ of positive degree in $x_{1}$. As in $\S 5$, we write \begin{aligned} &f=a_{0} x_{1}^{\prime}+\cdots+a_{l}, \quad a_{0} \neq 0 \ &g=b_{0} x_{1}^{m}+\cdots+b_{m}, \quad b_{0} \neq 0, \end{aligned} where $a_{i}, b_{i} \in k\left[x_{2}, \ldots, x_{n}\right]$, and we define the resultant of $f$ and $g$ with respect to $x_{1}$

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions