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## 数学代写|抽象代数代写Abstract Algebra代考|Theorems and proofs in Abstract Algebra

irst things first: if you flipped straight to this chapter because you are struggling with theorems and proofs in Abstract Algebra, please start with Chapter 2. People often work on theorems and proofs without fully understanding the underlying axioms and definitions, which makes everything more difficult. If you have read Chapter 2 , you will know that I think of mathematical theories as in the diagram below, where the ‘bottom’ layer contains axioms and definitions. Theorems are proved from these axioms and definitions; they are the ‘results’ of the deductive science that is mathematics.

But what does it mean to prove theorems or to say that mathematics is deductive? A deductive argument is one in which the conclusion is a necessary consequence of the premises (also called assumptions or hypotheses). You might not have thought of mathematics this way, but you are nevertheless accustomed to deductive reasoning. The algebraic step from ‘ $(x-2)(x-5)=0$ ‘ to ‘ $x=2$ or $x=5$ ‘ is a deduction: if the premise that $(x-2)(x-5)=0$ is true, it necessarily follows that $x=2$ or $x=5$. A proof chains such deductions together: each step introduces relevant objects or can be justified using axioms, definitions and earlier results. Proofs can be long and complicated, but they can also be short and simple, like this.

## 数学代写|抽象代数代写Abstract Algebra代考|Logic in familiar algebra

This section reviews some familiar algebra with a focus on its underlying logic. You might read it and think, yeah, I know all that. But many readers will recognize that while they ‘know’ it in the sense that they could act accordingly, they have not systematically reflected upon this knowledge. This review will build on the discussion in Section 1.2, considering logic in algebra in relation to numbers and matrices.

First, consider equation solving. What would you say it means to solve an equation? Can you get beyond ‘finding $x$ ‘? Maybe think about how you would explain equation solving to a young student who is intelligent but has not yet studied equations. What would you say? I will ask again later. Perhaps the simplest equations take forms as in Section $1.3: x+a=b$ or $a x=b$. If these seem trivial, that is due to your extensive knowledge. For young children, the world of numbers is smaller than it is for you, and the equations $x+5=2$ and $5 x=2$ have no solutions. For you, they do have solutions because you know about negative and rational numbers. In Abstract Algebra, we do not revert to the earlier position, but nor do we assume that all numbers are always fair game-we are careful about sets. In the integers, $5 x=2$ has no solution. In the real numbers, $x^2=-5$ has no solution.

As discussed in Section 1.3, to guarantee that all equations of the form $x+a=b$ can be solved requires that the manipulations below are valid. So it requires a set that is closed under addition and that has associative addition, an additive identity and additive inverses. In short, it requires an additive group.

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|What is abstract about Abstract Algebra?

Abstract Algebra is abstract in the same sense in which other Auman thinking is abstract: its concepts can be instantiated in ple ways. For instance, you recognize things like trees and You can do that because you understand the abstract ideas ‘tree’ ‘window’, and you can match them to objects in the world. You do not need to look at one tree to identify another; you do it by reference to the abstract idea.

Now, trees and windows are physical objects-you can walk up and touch them, and identify them by sight. But you can think about concepts that are more abstract, too. For instance, you can identify an aunt. You do that by reference to a criterion: is this a female person with a sibling who has children? If yes, it’s an aunt. If no, it’s not. Moreover, you can think about abstract concepts that are not single objects, like family. A family includes multiple people-perhaps many-who are related to one another-genetically or by marriage or by other caring relationships. Families vary a lot, and you couldn’t necessarily recognize a family by sight or by checking simple criteria. But you nevertheless understand the idea.

Abstract Algebra is about concepts that are somewhat like each of these more abstract ideas. They are like aunts in that they are defined by criteria. Abstract Algebra is stricter, though. Everyday human concepts, even defined ones like ‘aunt’, tend to be used flexibly. Is your mum’s brother’s female partner your aunt? Maybe, maybe not. And where I grew up, adult female neighbours and friends were commonly referred to as ‘Auntie’, even where there were no family relationships. Such flexibility doesn’t happen in Abstract Algebra, because mathematical concepts are specified by precise definitions about which all mathematicians agree. ${ }^1$

## 数学代写|抽象代数代写Abstract Algebra代考|What is algebraic about Abstract Algebra?

To understand what is algebraic about Abstract Algebra, it is probably useful to consider what is algebraic about earlier algebra. Many students think of algebra as something you do, where doing algebra means manipulating an expression or equation in valid ways to arrive at another. This is often in service of a goal: solving a mechanics problem, say. And thinking of algebra in this way is not wrong-certainly it captures most students’ experience prior to undergraduate mathematics. But it is not enough to grasp the aims of Abstract Algebra.

Abstract Algebra focuses not on performing algebraic manipulations but on understanding the mathematical structures that make those manipulations valid. To see what I mean, consider this algebraic argument (the arrow ‘ $\Rightarrow$ ‘ means ‘implies’).
\begin{aligned} & x(x+y)=y x \ & \Rightarrow x^2+x y-y x=0 \ & \Rightarrow \quad x^2=0 \ & \Rightarrow \quad x=0 \text {. } \ & \end{aligned}

## 数学代写|抽象代数代写Abstract Algebra代考|What is algebraic about Abstract Algebra?

$$x(x+y)=y x \quad \Rightarrow x^2+x y-y x=0 \Rightarrow \quad x^2=0 \quad \Rightarrow \quad x=0$$

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Coset Decoding

There is another convenient decoding method that utilizes the fact that an $(n, k)$ linear code $C$ over a finite field $F$ is a subgroup of the additive group of $V=F^n$ . This method was devised by David Slepian in 1956 and is called coset decoding (or standard decoding). To use this method, we proceed by constructing a table, called a standard array. The first row of the table is the set $C$ of code words, beginning in column 1 with the identity $0 \cdots 0$. To form additional rows of the table, choose an element $v$ of $V$ not listed in the table thus far. Among all the elements of the coset $v+C$, choose one of minimum weight, say, v’. Complete the next row of the table by placing under the column headed by the code word $c$ the vector $v^{\prime}+c$. Continue this process until all the vectors in $V$ have been listed in the table. [Note that an $(n, k)$ linear code over a field with $q$ elements will have $|V: C|=q^{n-k}$ rows.] The words in the first column are called the coset leaders. The decoding procedure is simply to decode any received word $w$ as the code word at the head of the column containing $w$.

I EXAMPLE 11 Consider the $(6,3)$ binary linear code
$$C={000000,100110,010101,001011,110011,101101,011110,111000} .$$
The first row of a standard array is just the elements of $C$. Obviously, 100000 is not in $C$ and has minimum weight among the elements of $100000+C$, so it can be used to lead the second row. Table 29.4 is the completed table.

## 数学代写|抽象代数代写Abstract Algebra代考|Historical Note

In this “Age of Information,” no one need be reminded of the importance not only of speed but also of accuracy in the storage, retrieval, and transmission of data. Machines do make errors, and their non-man-made mistakes can turn otherwise flawless programming into worthless, even dangerous, trash. Just as architects design buildings that will remain standing even through an earthquake, their computer counterparts have come up with sophisticated techniques capable of counteracting digital disasters.

The idea for the current error-correcting techniques for everything from computer hard disk drives to CD players was first introduced in 1960 by Irving Reed and Gustave Solomon, then staff members at MIT’s Lincoln Laboratory….
“When you talk about CD players and digital audio tape and now digital television, and various other digital imaging systems that are coming-all of those need Reed-Solomon [codes] as an integral part of the system,” says Robert McEliece, a coding theorist in the electrical engineering department at Caltech.

Why? Because digital information, virtually by definition, consists of strings of “bits”-0’s and 1’s – and a physical device, no matter how capably manufactured, may occasionally confuse the two. Voyager II, for example, was transmitting data at incredibly low power-barely a whisper — over tens of millions of miles. Disk drives pack data so densely that a read/write head can (almost) be excused if it can’t tell where one bit stops and the next 1 (or 0 ) begins. Careful engineering can reduce the error rate to what may sound like a negligible level – the industry standard for hard disk drives is 1 in 10 billionbut given the volume of information processing done these days, that “negligible” level is an invitation to daily disaster. Error-correcting codes are a kind of safety net-mathematical insurance against the vagaries of an imperfect material world.
In 1960, the theory of error-correcting codes was only about a decade old. The basic theory of reliable digital communication had been set forth by Claude Shannon in the late 1940s. At the same time, Richard Hamming introduced an elegant approach to single-error correction and double-error detection. Through the 1950s, a number of researchers began experimenting with a variety of errorcorrecting codes. But with their SIAM journal paper, McEliece says, Reed and Solomon “hit the jackpot.”

## 数学代写|抽象代数代写Abstract Algebra代考|Coset Decoding

$$C=000000,100110,010101,001011,110011,101101,011110,111000 .$$

## 数学代写|抽象代数代写Abstract Algebra代考|Historical Note

1960 年，欧文·里德 (Irving Reed) 和古斯塔夫·所罗门 (Gustave Solomon) 首次提出了从计算机硬盘驱动器到 CD 播放器的当前纠错技术的想法，然后是麻省理工学院林肯实验室的工作人员……。
“当你谈论 CD 播放器和数字音频磁带以及现在的数字电视和即将到来的各种其他数字成像系统时，所有这些都需要 Reed-Solomon [代码] 作为系统的组成部分，”Robert McEliece 说，他是一名加州理工学院电气工程系的编码理论家。

1960 年，纠错码理论只有大约十年的历史。克劳德·香农 (Claude Shannon) 在 20 世纪 40 年代后期提出了可靠数字通信的基本理论。与此同时，Richard Hamming 引入了一种优雅的单错误纠正和双错误检测方法。整个 1950 年代，许多研究人员开始试验各种纠错码。但 McEliece 说，凭借他们的 SIAM 期刊论文，里德和所罗门“中了大奖”。

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Abelian Groups Have Hamiltonian Paths

Let $G$ be a finite Abelian group, and let $S$ be any generating set for $G$. Then Cay(S:G) has a Hamiltonian path.

PROOF We use induction on $|S|$. If $|S|=1$, say, $S={a}$, then the digraph is just a circle labeled with $e, a, a^2, \ldots, a^{m-1}$, where $|a|=m$. Obviously, there is a Hamiltonian path for this case. Now assume that $|S|>1$. Choose some $s \in S$. Let $T=S-{s}$ – that is, $T$ is $S$ with $s$ removed-and set $H=\left\langle s_1, s_2, \ldots, s_{n-1}\right\rangle$ where $S=\left{s_1, s_2, \ldots, s_n\right}$ and $s=s_n$. (Notice that $H$ may be equal to $G$.)
Because $|T|<|S|$ and $H$ is a finite Abelian group, the induction hypothesis guarantees that there is a Hamiltonian path $\left(a_1, a_2, \ldots, a_k\right)$ in $\operatorname{Cay}(T: H)$. We will show that
$$\left(a_1, a_2, \ldots, a_k, s, a_1, a_2, \ldots, a_k, s, \ldots, a_1, a_2, \ldots, a_k, s, a_1, a_2, \ldots, a_k\right) \text {, }$$
where $a_1, a_2, \ldots, a_k$ occurs $|G| /|H|$ times and $s$ occurs $|G| /|H|-1$ times, is a Hamiltonian path in $\operatorname{Cay}(S: G)$.

## 数学代写|抽象代数代写Abstract Algebra代考|Some Applications

Cayley digraphs are natural models for interconnection networks in computer designs, and Hamiltonicity is an important property in relation to sorting algorithms on such networks. One particular Cayley digraph that is used to design and analyze interconnection networks of parallel machines is the symmetric group $S_n$ with the set of all transpositions as the generating set. Hamiltonian paths and circuits in Cayley digraphs arise in a variety of group theory contexts. A Hamiltonian path in a Cayley digraph of a group is simply an ordered listing of the group elements without repetition. The vertices of the digraph are the group elements, and the arcs of the path are generators of the group. In 1948, R. A. Rankin used these ideas (although not the terminology) to prove that certain bell-ringing exercises could not be done by the traditional methods employed by bell ringers. In 1981, Hamiltonian paths in Cayley digraphs were used in an algorithm for creating computer graphics of Escher-type repeating patterns in the hyperbolic plane. This program can produce repeating hyperbolic patterns in color from among various infinite classes of symmetry groups. The program has now been improved so that the user may choose from many kinds of color symmetry. The 2003 Mathematics Awareness Month poster featured one such image (see mathaware.org/mam/03). Two Escher drawings and their computer-drawn counterparts are given in Figures 28.9 through 28.12.

## 数学代写|抽象代数代写Abstract Algebra代考|Abelian Groups Have Hamiltonian Paths

$$\left(a_1, a_2, \ldots, a_k, s, a_1, a_2, \ldots, a_k, s, \ldots, a_1, a_2, \ldots, a_k, s, a_1, a_2, \ldots, a_k\right),$$

## 数学代写|抽象代数代写Abstract Algebra代考|Some Applications

Cayley 有向图是计算机设计中互连网络的自然模型，而哈密顿性是与此类网络上的排序算法相关的重要属性。一种用于设计和分析并行机互连网络的特定凯莱有向图是对称群

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Classification of Groups of Order Up to 15

The next theorem illustrates the utility of the ideas presented in this chapter.
Theorem 25.4 Classification of Groups of Order 8 (Cayley, 1859)
Up to isomorphism, there are only five groups of order 8 :
$Z_8, Z_4 \oplus Z_2, Z_2 \oplus Z_2 \oplus Z_2, D_4$, and the quaternions.

PROOF The Fundamental Theorem of Finite Abelian Groups takes care of the Abelian cases. Now, let $G$ be a non-Abelian group of order 8 . Also, let $G_1=\left\langle a, b \mid a^4=b^2=(a b)^2=e\right\rangle$ and let $G_2=\left\langle a, b \mid a^2=b^2=(a b)^2\right\rangle$. We know from the preceding examples that $G_1$ is isomorphic to $D_4$ and $G_2$ is isomorphic to the quaternions. Thus, it suffices to show that $G$ must satisfy the defining relations for $G_1$ or $G_2$. It follows from Exercise 45 in Chapter 2 and Lagrange’s Theorem that $G$ has an element of order 4 ; call it $a$. Then, if $b$ is any element of $G$ not in $\langle a\rangle$, we know that
$$G=\langle a\rangle \cup\langle a\rangle b=\left{e, a, a^2, a^3, b, a b, a^2 b, a^3 b\right} .$$

## 数学代写|抽象代数代写Abstract Algebra代考|Characterization of Dihedral Groups

As another nice application of generators and relations, we will now give a characterization of the dihedral groups that has been known for more than 100 years. For $n \geq 3$, we have used $D_n$ to denote the group of symmetries of a regular $n$-gon. Imitating Example 2, one can show that
$D_n \approx\left\langle a, b \mid a^n=b^2=(a b)^2=e\right\rangle$ (see Exercise 9). By analogy, these generators and relations serve to define $D_1$ and $D_2$ also. (These are also called dihedral groups.) Finally, we define the infinite dihedral group $D_{\infty}$ as $\left\langle a, b \mid a^2=b^2=e\right\rangle$. The elements of $D_{\infty}$ can be listed as $e, a, b, a b, b a,(a b) a,(b a) b,(a b)^2,(b a)^2,(a b)^2 a,(b a)^2 b,(a b)^3,(b a)^3, \ldots$
Theorem 25.5. Characterization of Dihedral Groups
Any group generated by a pair of elements of order 2 is dihedral.
PROOF Let $G$ be a group generated by a pair of distinct elements of order 2 , say, $a$ and $b$. We consider the order of $a b$. If $|a b|=\infty$, then $G$ is infinite and satisfies the relations of $D_{\infty}$. We will show that $G$ is isomorphic to $D_{\infty}$. By Dyck’s Theorem, $G$ is isomorphic to some factor group of $D_{\infty}$, say, $D_{\infty} / H$. Now, suppose $h \in H$ and $h \neq e$. Since every element of $D_{\infty}$ has one of the forms $(a b)^i,(b a)^i,(a b)^i a$, or $(b a)^i b$, by symmetry, we may assume that $h=(a b)^i$ or $h=(a b)^i a$. If $h=(a b)^i$, we will show that $D_{\infty} / H$ satisfies the relations for $D_i$ given in Exercise 9. Since $(a b)^i$ is in $H$, we have
$$H=(a b)^i H=(a b H)^i$$

so that $(a b H)^{-1}=(a b H)^{i-1}$. But
$$(a b)^{-1} H=b^{-1} a^{-1} H=b a H,$$
and it follows that
$$a H a b H a H=a^2 H b H a H=e H b a H=b a H=(a b H)^{-1} .$$

## 数学代写|抽象代数代写Abstract Algebra代考|Classification of Groups of Order Up to 15

$Z_8, Z_4 \oplus Z_2, Z_2 \oplus Z_2 \oplus Z_2, D_4$ 和四元数。

left 缺少或无法识别的分隔符

## 数学代写|抽象代数代写Abstract Algebra代考|Characterization of Dihedral Groups

$D_n \approx\left\langle a, b \mid a^n=b^2=(a b)^2=e\right\rangle$ (见练习9) 。以此类推，这些生成器和关系用于定义 $D_1$ 和 $D_2$ 还。 (这些也称为二面角群。) 最后，我们定义无限二面角群 $D_{\infty}$ 作为 $\left\langle a, b \mid a^2=b^2=e\right\rangle$. 的 元素 $D_{\infty}$ 可以列为 $e, a, b, a b, b a,(a b) a,(b a) b,(a b)^2,(b a)^2,(a b)^2 a,(b a)^2 b,(a b)^3,(b a)^3, \ldots$ 定理 25.5。二面角群的特征

$(a b)^i,(b a)^i,(a b)^i a$ ，或者 $(b a)^i b$ ，通过对称性，我们可以假设 $h=(a b)^i$ 或者 $h=(a b)^i a$. 如果
$$H=(a b)^i H=(a b H)^i$$

$$(a b)^{-1} H=b^{-1} a^{-1} H=b a H,$$

$$a H a b H a H=a^2 H b H a H=e H b a H=b a H=(a b H)^{-1} .$$

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Applications of Sylow Theorems

Theorem $21.1$ gives us a complete description of all finite fields. The following theorem gives us a complete description of all the subfields of a finite field. Notice the close analogy between this theorem and Theorem 4.3, which describes all the subgroups of a finite cyclic group.
Theorem 21.11 Subfields of a Finite Field
For each divisor $m$ of $n, \operatorname{GF}\left(p^n\right)$ has a unique subfield of order $p^m$. Moreover, these are the only subfields of $\mathrm{GF}\left(p^n\right)$.

PROOF To show the existence portion of the theorem, suppose that $m$ divides $n$. Then, since
$$p^n-1=\left(p^m-1\right)\left(p^{n-m}+p^{n-2 m}+\cdots+p^m+1\right),$$
we see that $p^m-1$ divides $p^n-1$. For simplicity, write $p^n-1=\left(p^m-1\right) t$. Let $K=\left{x \in \operatorname{GF}\left(p^n\right) \mid x^{p^m}=x\right}$. We leave it as an easy exercise for the reader to show that $K$ is a subfield of $\operatorname{GF}\left(p^n\right)$ (Exercise 37). Since the polynomial $x^{p^m}-x$ has at most $p^m$ zeros in $\operatorname{GF}\left(p^n\right)$, we have $|K| \leq p^m$. Let $\langle a\rangle=\operatorname{GF}\left(p^n\right)^*$. Then $\left|a^t\right|=p^m-1$, and since $\left(a^t\right)^{p^{m-1}}=1$ , it follows that $a^t \in K$. So, $K$ is a subfield of $\operatorname{GF}\left(p^n\right)$ of order $p^m$.

The uniqueness portion of the theorem follows from the observation that if $\operatorname{GF}\left(p^n\right)$ had two distinct subfields of order $p^m$, then the polynomial $x^{p^m}-x$ would have more than $p^m$ zeros in $\operatorname{GF}\left(p^n\right)$. This contradicts Theorem 16.3.

## 数学代写|抽象代数代写Abstract Algebra代考|Cyclic Groups of Order $p q$

If $G$ is a group of order $p q$, where $p$ and $q$ are primes, $p<q$, and $p$ does not divide $q-1$, then $G$ is cyclic. In particular, $G$ is isomorphic to $Z_{p q}$.

PROOF Let $H$ be a Sylow $p$-subgroup of $G$ and let $K$ be a Sylow $q$-subgroup of $G$. Sylow’s Third Theorem states that the number of Sylow $p$-subgroups of $G$ is of the form $1+k p$ and divides $q$. So $1+k p=1$ or $1+k p=q$. Since $p$ does not divide $q-1$, we have that $k=0$ and therefore $H$ is the only Sylow $p$-subgroup of $G$.

Similarly, there is only one Sylow $q$-subgroup of $G$ (see Exercise 25). Thus, by the corollary to Theorem 23.5, $H$ and $K$ are normal subgroups of $G$. Moreover, from Theorem $7.2$ and Lagrange, we have $G=H K$ and $H \cap K={e}$. This tells us that $G=H \times K$. Finally, by Theorem $8.2, G \approx Z_p \oplus Z_q \approx Z_{p q}$.

Theorem $23.6$ demonstrates the power of the Sylow theorems in classifying the finite groups whose orders have small numbers of prime factors. Similar results exist for groups of orders $p^2 q, p^2 q^2, p^3$, and $p^4$, where $p$ and $q$ are prime.

For your amusement, Figure $23.2$ lists the number of nonisomorphic groups with order at most 100. Note in particular the large number of groups of order 64. Also observe that, generally speaking, it is not the size of the group that gives rise to a large number of groups of that size but the number of prime factors involved. In all, there are 1047 nonisomorphic groups with 100 or fewer elements. Contrast this with the fact that there are 49,487,365,422 groups of order $1024=2^{10}$. The number of groups of any order less than 2048 is given at http://oeis.org/A000001/b000001.txt.

## 数学代写|抽象代数代写Abstract Algebra代考|Applications of Sylow Theorems

$$p^n-1=\left(p^m-1\right)\left(p^{n-m}+p^{n-2 m}+\cdots+p^m+1\right),$$

\left 缺少或无法识别的分隔符 $\quad$. 我们把它作为一个简单的练习留给读者来证明 $K$ 是一个子 字段 $G F\left(p^n\right)$ (练习37) 。由于多项式 $x^{p^m}-x$ 至多有 $p^m$ 归霝 $\mathrm{GF}\left(p^n\right)$ ，我们有 $|K| \leq p^m$. 让
$\langle a\rangle=\operatorname{GF}\left(p^n\right)^*$. 然后 $\left|a^t\right|=p^m-1$ ，并且因为 $\left(a^t\right)^{p^{m-1}}=1$ ，它逽循 $a^t \in K$. 所以， $K$ 是一个子字段 $\operatorname{GF}\left(p^n\right)$ 秩序 $p^m$.

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Subfields of a Finite Field

Theorem $21.1$ gives us a complete description of all finite fields. The following theorem gives us a complete description of all the subfields of a finite field. Notice the close analogy between this theorem and Theorem 4.3, which describes all the subgroups of a finite cyclic group.
Theorem 21.11 Subfields of a Finite Field
For each divisor $m$ of $n, \operatorname{GF}\left(p^n\right)$ has a unique subfield of order $p^m$. Moreover, these are the only subfields of $\mathrm{GF}\left(p^n\right)$.

PROOF To show the existence portion of the theorem, suppose that $m$ divides $n$. Then, since
$$p^n-1=\left(p^m-1\right)\left(p^{n-m}+p^{n-2 m}+\cdots+p^m+1\right),$$
we see that $p^m-1$ divides $p^n-1$. For simplicity, write $p^n-1=\left(p^m-1\right) t$. Let $K=\left{x \in \operatorname{GF}\left(p^n\right) \mid x^{p^m}=x\right}$. We leave it as an easy exercise for the reader to show that $K$ is a subfield of $\operatorname{GF}\left(p^n\right)$ (Exercise 37). Since the polynomial $x^{p^m}-x$ has at most $p^m$ zeros in $\operatorname{GF}\left(p^n\right)$, we have $|K| \leq p^m$. Let $\langle a\rangle=\operatorname{GF}\left(p^n\right)^*$. Then $\left|a^t\right|=p^m-1$, and since $\left(a^t\right)^{p^{m-1}}=1$ , it follows that $a^t \in K$. So, $K$ is a subfield of $\operatorname{GF}\left(p^n\right)$ of order $p^m$.

The uniqueness portion of the theorem follows from the observation that if $\operatorname{GF}\left(p^n\right)$ had two distinct subfields of order $p^m$, then the polynomial $x^{p^m}-x$ would have more than $p^m$ zeros in $\operatorname{GF}\left(p^n\right)$. This contradicts Theorem 16.3.

## 数学代写|抽象代数代写Abstract Algebra代考|Historical Discussion of Geometric Constructions

The ancient Greeks were fond of geometric constructions. They were especially interested in constructions that could be achieved using only a straightedge without markings and a compass. They knew, for example, that any angle can be bisected, and they knew how to construct an equilateral triangle, a square, a regular pentagon, and a regular hexagon. But they did not know how to trisect every angle or how to construct a regular seven-sided polygon (heptagon). Another problem that they attempted was the duplication of the cube-that is, given any cube, they tried to construct a new cube having twice the volume of the given one using only an unmarked straightedge and a compass. Legend has it that the ancient Athenians were told by the oracle at Delos that a plague would end if they constructed a new altar to Apollo in the shape of a cube with double the volume of the old altar, which was also a cube. Besides “doubling the cube,” the Greeks also attempted to “square the circle”- to construct a square with area equal to that of a given circle. They knew how to solve all these problems using other means, such as a compass and a straightedge with two marks, or an unmarked straightedge and a spiral, but they could not achieve any of the constructions with a compass and an unmarked straightedge alone. These problems vexed mathematicians for over 2000 years. The resolution of these perplexities was made possible when they were transferred from questions of geometry to questions of algebra in the 19 th century.
The first of the famous problems of antiquity to be solved was that of the construction of regular polygons. It had been known since Euclid that regular polygons with a number of sides of the form $2^k, 2^k \cdot 3,2^k \cdot 5$, and $2^k \cdot 3 \cdot 5$ could be constructed, and it was believed that no others were possible. In 1796, while still a teenager, Gauss proved that the 17-sided regular polygon is constructible. In 1801, Gaussbio]Gauss, Carl asserted that a regular polygon of $n$ sides is constructible if and only if $n$ has the form $2^k p_1 p_2 \cdots p_i$, where the $p$ ‘s are distinct primes of the form $2^{2^s}+1$. We provide a proof of this statement in Theorem 31.5.

## 数学代写|抽象代数代写Abstract Algebra代考|Subfields of a Finite Field

$$p^n-1=\left(p^m-1\right)\left(p^{n-m}+p^{n-2 m}+\cdots+p^m+1\right),$$

\left 缺少或无法识别的分隔符 . 我们把它作为一个简单的练习留给读者来证明 $K$ 是一个子 字段 $G F\left(p^n\right)$ (练习 37) 。由于多项式 $x^{p^m}-x$ 至多有 $p^m$ 归零 $\mathrm{GF}\left(p^n\right)$ ，我们有 $|K| \leq p^m$. 让
$\langle a\rangle=\operatorname{GF}\left(p^n\right)^*$. 然后 $\left|a^t\right|=p^m-1$ ，并且因为 $\left(a^t\right)^{p^{m-1}}=1$ ，它遭循 $a^t \in K$. 所以， $K$ 是一个子字段 $\mathrm{GF}\left(p^n\right)$ 秩序 $p^m$. $\mathrm{GF}\left(p^n\right)$. 这与定理 $16.3$ 相矛盾。

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Maximality of ⟨𝑔(𝑥)⟩

Example $36.8$ (Non-example). Consider $x^2-1 \in \mathbb{R}[x]$. We’ll show that the ideal $\left\langle x^2-1\right\rangle$ is not maximal in $\mathbb{R}[x]$ by finding an ideal $A$ that is strictly between $\left\langle x^2-1\right\rangle$ and $\mathbb{R}[x]$; i.e., $\left\langle x^2-1\right\rangle \subsetneq A \subsetneq \mathbb{R}[x]$. Noting that $x^2-1$ factors as $x^2-1=(x+1) \cdot(x-1)$, we’ll let $A=\langle x+1\rangle$.

First, we’ll show that $\left\langle x^2-1\right\rangle \subsetneq A$. If $\alpha(x) \in\left\langle x^2-1\right\rangle$, then $\alpha(x)=\left(x^2-1\right) \cdot q(x)$ for some $q(x) \in \mathbb{R}[x]$. Then $\alpha(x)=((x+1) \cdot(x-1)) \cdot q(x)=(x+1) \cdot((x-1) \cdot q(x))$, so that $\alpha(x)$ is a multiple of $x+1$; i.e., $\alpha(x) \in A$. Hence, $\left\langle x^2-1\right\rangle \subseteq A$. Moreover, $x+1$ is a multiple of $x+1$, but not a multiple of $x^2-1$. Therefore, $x+1 \in A$, but $x+1 \notin\left\langle x^2-1\right\rangle$. This show that $\left\langle x^2-1\right\rangle \neq A$, so that $\left\langle x^2-1\right\rangle \subsetneq A$.

Certainly, $A \subseteq \mathbb{R}[x]$. But $A \neq \mathbb{R}[x]$, as $1 \in \mathbb{R}[x]$ but $1 \notin A$. (Note that the only constant polynomial in $A=\langle x+1\rangle$ is 0.) Thus, we obtain $A \subsetneq \mathbb{R}[x]$. Therefore, we find that $\left\langle x^2-1\right\rangle \subsetneq A \subsetneq \mathbb{R}[x]$, so that $A=\langle x+1\rangle$ is an ideal that is strictly between $\left\langle x^2-1\right\rangle$ and $\mathbb{R}[x]$. We conclude that $\left\langle x^2-1\right\rangle$ is not maximal. The picture below illustrates these set inclusions:

Proof know-how. To show that $M \subsetneq A$, we must show that $M \subseteq A$ and $M \neq A$. Showing $M \subseteq A$ can be done in the familiar manner: Consider an element $m \in M$ and show that $m \in A$. To show that $M \neq A$ (i.e., $M$ and $A$ are different sets), one approach is to find an element $a \in A$ such that $a \notin M$. In Example 36.8, for instance, we showed that $\left\langle x^2-1\right\rangle \neq A$ by showing that $x+1 \in A$, but $x+1 \notin\left\langle x^2-1\right\rangle$.

## 数学代写|抽象代数代写Abstract Algebra代考|Big picture stuff

By using the contrapositive of its first statement, Theorem $36.14$ can be restated as:
$$\langle g(x)\rangle \text { is maximal in } F[x] \text { if and only if } g(x) \text { is unfactorable. }$$
Exercise $# 5$ at the end of this chapter is about the ideal $\langle n\rangle$ (or $n \mathbb{Z}$ ) in $\mathbb{Z}$. It states the following:
$\langle n\rangle$ is maximal in $\mathbb{Z}$ if and only if $n$ is prime.
Notice how these two statements are essentially the same. This is yet another instance of the structural similarity between the ring of integers $\mathbb{Z}$ and the polynomial ring $F[x]$ where $F$ is a field.

## 数学代写|抽象代数代写Abstract Algebra代考|Maximality of $\langle g(x)\rangle$

$A=\langle x+1\rangle$ 为 0 。) 因此，我们得到 $A \subsetneq \mathbb{R}[x]$. 因此，我们发现 $\left\langle x^2-1\right\rangle \subsetneq A \subsetneq \mathbb{R}[x]$, 以便 $A=\langle x+1\rangle$ 是严格介于两者之间的理想 $\left\langle x^2-1\right\rangle$ 和 $\mathbb{R}[x]$. 我们的结论是 $\left\langle x^2-1\right\rangle$ 不是最大的。下 图说明了这些套装内含物:

## 数学代写|抽象代数代写Abstract Algebra代考|Big picture stuff

$\langle g(x)\rangle$ is maximal in $F[x]$ if and only if $g(x)$ is unfactorable.

$\langle n\rangle$ 是最大的 $\mathbb{Z}$ 当且仅当 $n$ 是质数。

## 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。

Posted on Categories:abstract algebra, 抽象代数, 数学代写

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## 数学代写|抽象代数代写Abstract Algebra代考|𝐹[𝑥]/⟨𝑔(𝑥)⟩ is not a field

Thus far, we have seen two examples of quotient rings involving polynomials:

• $\mathbb{Z}_3[x] /\left\langle x^2\right\rangle$ in Section $32.3$.
• $\mathbb{Z}_7[x] /\left\langle x^2-1\right\rangle$ in this chapter.
Each of these has a zero divisor and thus is not a field. Moreover, we found these zero divisors by factoring. With $\mathbb{Z}_3[x] /\left\langle x^2\right\rangle$, we consider the factorization $x^2=x \cdot x$. The element $x+\left\langle x^2\right\rangle \in \mathbb{Z}_3[x] /\left\langle x^2\right\rangle$ is non-zero, because $x$ is not a multiple of $x^2$. And we have
$$\left(x+\left\langle x^2\right\rangle\right) \cdot\left(x+\left\langle x^2\right\rangle\right)=x^2+\left\langle x^2\right\rangle=0+\left\langle x^2\right\rangle,$$
so that $x+\left\langle x^2\right\rangle$ is a zero divisor in $\mathbb{Z}_3[x] /\left\langle x^2\right\rangle$. With $\mathbb{Z}_7[x] /\left\langle x^2-1\right\rangle$, we factor $x^2-1=$ $(x+1) \cdot(x-1)$ and proceed similarly to find zero divisors $(x+1)+\left\langle x^2-1\right\rangle$ and $(x-1)+\left\langle x^2-1\right\rangle$. Here is the generalization, whose proof is left for you as an exercise.

## 数学代写|抽象代数代写Abstract Algebra代考|𝐹[𝑥]/⟨𝑔(𝑥)⟩ is a field

Based on Theorem 33.8, we might conjecture the following:
Let $F$ be a field and fix $g(x) \in F[x]$. If $g(x)$ is unfactorable, then $F[x] /\langle g(x)\rangle$ is $a$ field.
Below are some examples that support the conjecture.
Example 33.10. Consider the polynomial $g(x)=x^2-2$, which is unfactorable in $\mathbb{Q}[x]$. (See Example 30.21.) Do we also know that the quotient ring $\mathbb{Q}[x] /\left\langle x^2-2\right\rangle$ is a field? In Example 32.13, we used the First Isomorphism Theorem to derive a ring isomorphism $\mathbb{Q}[x] /\left\langle x^2-2\right\rangle \cong \mathbb{Q}(\sqrt{2})$. Moreover, $\mathbb{Q}(\sqrt{2})$ is a field. (See Chapter 27, Exercise #14(c).) Therefore $\mathbb{Q}[x] /\left\langle x^2-2\right\rangle$ is a field, as desired.

Example 33.11. Consider $g(x)=x^2+1 \in \mathbb{Z}_3[x]$. We have $g(0)=1, g(1)=2$, and $g(2)=2$, so that $g(x)$ does not have a root in $\mathbb{Z}_3$. And since $\operatorname{deg} g(x)=2$, we conclude that $g(x)$ is unfactorable in $\mathbb{Z}_3[x]$ by Theorem $30.19$. In the exercises, you’ll verify that the quotient ring $\mathbb{Z}_3[x] /\left\langle x^2+1\right\rangle$ is indeed a field.

Example 33.12. Consider $g(x)=x^2+1 \in \mathbb{Z}_7[x]$. In the exercises, you’ll verify the following:

• $g(x)$ does not have a root in $\mathbb{Z}_7$, so that $g(x)$ is unfactorable in $\mathbb{Z}_7[x]$.
• The quotient ring $\mathbb{Z}_7[x] /\left\langle x^2+1\right\rangle$ is a field.
Hence this example also supports our conjecture.

## 数学代写|抽象代数代写Abstract Algebra代考 $\mid F[x] /\langle g(x)\rangle$ is not a field

• $\mathbb{Z}_3[x] /\left\langle x^2\right\rangle$ 在部分 $32.3$.
• $\mathbb{Z}_7[x] /\left\langle x^2-1\right\rangle$ 在这一章当中。
其中每一个都有一个零除数，因此不是一个字段。此外，我们通过因式分解找到了这些零因 子。和 $\mathbb{Z}_3[x] /\left\langle x^2\right\rangle$ ，我们考虑因式分解 $x^2=x \cdot x$. 元素 $x+\left\langle x^2\right\rangle \in \mathbb{Z}_3[x] /\left\langle x^2\right\rangle$ 是非零 的，因为 $x$ 不是的倍数 $x^2$. 我们有
$$\left(x+\left\langle x^2\right\rangle\right) \cdot\left(x+\left\langle x^2\right\rangle\right)=x^2+\left\langle x^2\right\rangle=0+\left\langle x^2\right\rangle,$$
以便 $x+\left\langle x^2\right\rangle$ 是一个零因子 $\mathbb{Z}_3[x] /\left\langle x^2\right\rangle$. 和 $\mathbb{Z}_7[x] /\left\langle x^2-1\right\rangle$, 我们考虑 $x^2-1=$ $(x+1) \cdot(x-1)$ 并以类似的方式找到零除数 $(x+1)+\left\langle x^2-1\right\rangle$ 和 $(x-1)+\left\langle x^2-1\right\rangle$. 这是概括，其证明留给您作为练习。

## 数学代写|抽象代数代写Abstract Algebra代考 $\mid F[x] /\langle g(x)\rangle$ is a field

$\mathbb{Q}[x] /\left\langle x^2-2\right\rangle \cong \mathbb{Q}(\sqrt{2})$. 而且， $\mathbb{Q}(\sqrt{2})$ 是一个字段。（参见第 27 章，练习#14(c)。) 因此 $\mathbb{Q}[x] /\left\langle x^2-2\right\rangle$ 是一个字段，根据需要。

• $g(x)$ 没有根 $\mathbb{Z}_7$, 以便 $g(x)$ 在 $\mathbb{Z}_7[x]$.
• 商环 $\mathbb{Z}_7[x] /\left\langle x^2+1\right\rangle$ 是一个字段。 因此这个例子也支持了我们的猜想。

## 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。

Posted on Categories:abstract algebra, 抽象代数, 数学代写

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## 数学代写|抽象代数代写Abstract Algebra代考|Ideals in ℤ and in 𝐹[𝑥]

All of the ideals we’ve examined so far have been principal ideals of the form $\langle a\rangle=$ ${a \cdot r \mid r \in R}$, i.e., the set of all multiples of a fixed element $a \in R$. The next example shows that not every ideal is principal.

Example 31.31. Let $A$ be the set of all polynomials in $\mathbb{Z}[x]$ with an even constant term. For instance, consider $f(x)=3 x^{101}-171 x^{52}+x+12$ and $g(x)=5 x-21$, which are elements of $\mathbb{Z}[x]$. The constant terms of $f(x)$ and $g(x)$ are 12 (which is even) and $-21$ (which isn’t even), respectively. Therefore, $f(x) \in A$ and $g(x) \notin A$. In the exercises, you’ll show that (1) $A$ is an ideal of $\mathbb{Z}[x]$, but (2) $A$ is not a principal ideal; i.e., there does not exist an element $\alpha(x) \in \mathbb{Z}[x]$ such that $A=\langle\alpha(x)\rangle$.

Example 31.32. We’ve seen that $\langle 5\rangle=5 \mathbb{Z}$ is an ideal of $\mathbb{Z}$. Other ideals of $\mathbb{Z}$ include $\langle 2\rangle=2 \mathbb{Z},\langle 12\rangle=12 \mathbb{Z}$, and more generally, $\langle n\rangle=n \mathbb{Z}$, where $n$ is a fixed integer. These include the two extremes, i.e., the ring $\mathbb{Z}$ itself, $\langle 1\rangle=\mathbb{Z}$, and the trivial ideal $\langle 0\rangle={0}$.
It turns out that every ideal of $\mathbb{Z}$ is principal, as the following theorem states. We’ll start the proof but will leave it to you as an exercise to finish writing it.

## 数学代写|抽象代数代写Abstract Algebra代考|Big picture stuff

We continue to highlight the connections between the ring of integers $\mathbb{Z}$ and the polynomial ring $F[x]$, where $F$ is a field. As Theorems $31.33$ and $31.34$ indicate, both rings satisfy the condition that every ideal is principal. More generally, an integral domain whose ideals are all principal is called a principal ideal domain (or PID). And $\mathbb{Z}$ and $F[x]$ are classic examples of a PID.

For another connection, we restate an early observation in the language of principal ideals. In Chapter 3, Exercises #12 and #13, we proved the following:
Let $m, n \in \mathbb{Z}$. Then $n \mid m$ if and only if $\langle m\rangle \subseteq\langle n\rangle$.

We wrote $m \mathbb{Z}$ and $n \mathbb{Z}$ in Chapter 3 , but we saw in this chapter that those are equivalent to the principal ideals $\langle m\rangle$ and $\langle n\rangle$, respectively. Now, one of the exercises in this chapter states the following:
Let $f(x), g(x) \in F[x]$. Then $g(x) \mid f(x)$ if and only if $\langle f(x)\rangle \subseteq\langle g(x)\rangle$.
The actual proof is in $\mathbb{R}[x]$, but the argument remains the same in a more general setting of $F[x]$. Note how these statements are saying the same thing in two different rings $\mathbb{Z}$ and $F[x]$

## 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。