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# 数学代写|现代代数代考Modern Algebra代写|FACTORIZATION AND IDEALS

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## 数学代写|现代代数代考Modern Algebra代写|FACTORIZATION AND IDEALS

The theme in Chapter VIII was factorization; the theme in this chapter has been the mutually equivalent ideas of ring homomorphism, ideal, and quotient ring. In this section we shall see that these apparently unrelated themes are, in fact, not unrelated. Specifically, Theorem 41.2 will show that information about the ideals in a ring can often tell us when there is unique factorization in the ring. Then, by looking at some penetrating discoveries from nineteenth-century number theory, we shall see that in many rings factorization can most appropriately be studied by considering “products of ideals” in the ring, rather than just products of elements of the ring. This section is designed to show the relation between some general ideas connecting number theory and rings; details and proofs will be omitted or left to the problems. You should be familiar with Sections 37 and 38 , and Theorem 40.3. We begin with a key definition.

Definition. An integral domain in which every ideal is a principal ideal is called a principal ideal domain.

Examples of principal ideal domains include the ring of integers (Problem 38.17) and the ring $F[x]$ of polynomials over any field $F$ (Theorem 40.3 ). Recall that both $\mathbb{Z}$ and $F[x]$ are Euclidean domains; therefore, the fact that they are also principal ideal domains is a special case of the following theorem.
Theorem 41.1. Every Euclidean domain is a principal ideal domain.
The proof of Theorem 41.1 is similar to the proof of Theorem 40.3 , and will be left as an exercise (Problem 41.1). The converse of Theorem 41.1 is false; that is, not every principal ideal domain is a Euclidean domain. An example is given by the ring of all complex numbers of the form $a+b(1+\sqrt{-19}) / 2$ for $a, b \in \mathbb{Z}$. (See [5] for a discussion of this.)
The following theorem gives a direct link between factorization and ideals.

## 数学代写|现代代数代考Modern Algebra代写|SIMPLE EXTENSIONS. DEGREE

We begin by looking at how to construct field extensions that solve a particular kind of problem, namely that of providing roots for polynomials; the extension of $\mathbb{R}$ to $\mathbb{C}$ to obtain a root for $1+x^2$ (Section 32) is a special case.

Let $E$ be an extension field of a field $F$; for convenience, assume $F \subseteq E$. Also let $S$ be a subset of $E$. There is at least one subfield of $E$ containing both $F$ and $S$, namely $E$ itself. The intersection of all the subfields of $E$ that contain both $F$ and $S$ is a subfield of $E$ (Problem 42.1); it will be denoted $F(S)$. If $S \subseteq F$, then $F(S)=F$. If $S=\left{a_1, a_2, \ldots, a_n\right}$, then $F(S)$ will be denoted $F\left(a_1, a_2, \ldots, a_n\right)$. For example, $\mathbb{R}(i)=\mathbb{C}$. The field $F(S)$ consists of all the elements of $E$ that can be obtained from $F$ and $S$ by repeated applications of the operations of $E$-addition, multiplication, and the taking of additive and multiplicative inverses (Problem 42.3).

If $E=F(a)$ for some $a \in E$, then $E$ is said to be a simple extension of $F$. We can classify the simple extensions of $F$ by making use of $F[x]$, the ring of polynomials in the indeterminate $x$ over $F$, and
$$F[a]=\left{a_0+a_1 a+\cdots+a_n a^n: a_0, a_1, \ldots, a_n \in F\right},$$
the ring of all polynomials in $a$. The difference between $F[x]$ and $F[a]$ is that two polynomials in $F[x]$ are equal only if the coefficients on like powers of $x$ are equal, whereas if $a$ is algebraic over $F$ (Section 32), then two polynomials in $F[a]$ can be equal without the coefficients on like powers of $a$ being equal. For example,
$$1+3 \sqrt{2}=-1+3 \sqrt{2}+\sqrt{2}^2 \text { in } \mathbb{Q}[\sqrt{2}]$$
but
$$1+3 x \neq-1+3 x+x^2 \text { in } \mathbb{Q}[x]$$

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|SIMPLE EXTENSIONS. DEGREE

$$F[a]=\left{a_0+a_1 a+\cdots+a_n a^n: a_0, a_1, \ldots, a_n \in F\right},$$
$a$中所有多项式的环。$F[x]$和$F[a]$之间的区别在于，$F[x]$中的两个多项式只有在$x$的类似幂次上的系数相等时才相等，而如果$a$是$F$上的代数(第32节)，那么$F[a]$中的两个多项式可以相等，而$a$的类似幂次上的系数不相等。例如，
$$1+3 \sqrt{2}=-1+3 \sqrt{2}+\sqrt{2}^2 \text { in } \mathbb{Q}[\sqrt{2}]$$

$$1+3 x \neq-1+3 x+x^2 \text { in } \mathbb{Q}[x]$$

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## 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。