数学代写|抽象代数作业代写abstract algebra代考|The theory of equations and the Fundamental Theorem of Algebra

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抽象代数是代数的一组高级课题,涉及抽象代数结构而不是通常的数系。这些结构中最重要的是群、环和场。

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我们提供的抽象代数abstract algebra及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
  • Statistical Machine Learning 统计机器学习
  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础
数学代写|抽象代数作业代写abstract algebra代考|The theory of equations and the Fundamental Theorem of Algebra

数学代写|抽象代数作业代写abstract algebra代考|The theory of equations and the Fundamental Theorem of Algebra

Viète’s and Descartes’ work, in the late sixteenth and early seventeenth centuries, respectively, shifted the focus of attention from the solvability of numerical equations to theoretical studies of equations with literal coefficients. A theory of polynomial equations began to emerge. Among its main concerns were the determination of the existence, nature, and number of roots of such equations. Specifically:
(i) Does every polynomial equation have a root, and, if so, what kind of root is it? This was the most important and difficult of all questions on the subject. It turned out that the first part of the question was much easier to answer than the second. The Fundamental Theorem of Algebra (FTA) answered both: every polynomial equation, with real or complex coefficients, has a complex root.
(ii) How many roots does a polynomial equation have? In his Geometry, Descartes proved the Factor Theorem, namely that if $\alpha$ is a root of the polynomial $p(x)$, then $x-\alpha$ is a factor, that is, $p(x)=(x-\alpha) q(x)$, where $q(x)$ is a polynomial of degree

one less than that of $p(x)$. Repeating the process (formally, using induction), it follows that a polynomial of degree $n$ has exactly $n$ roots, given that it has one root, which is guaranteed by the FTA. The $n$ roots need not be distinct. This result means, then, that if $p(x)$ has degree $n$, there exist $n$ numbers $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ such that $p(x)=\left(x-\alpha_{1}\right)\left(x-\alpha_{2}\right) \ldots\left(x-\alpha_{n}\right)$. The FTA guarantees that the $\alpha_{i}$ are complex numbers. (Note that we speak interchangeably of the root $\alpha$ of a polynomial $p(x)$ and the root $\alpha$ of a polynomial equation $p(x)=0$; both mean $\left.p(\alpha)=0_{2}\right)$
(iii) Can we determine when the roots are rational, real, complex, positive? Every polynomial of odd degree with real coefficients has a real root. This was accepted on intuitive grounds in the seventeenth and eighteenth centuries and was formally established in the nineteenth as an easy consequence of the Intermediate Value Theorem in calculus, which says (in the version needed here) that a continuous function $f(x)$ which is positive for some values of $x$ and negative for others, must be zero for some $x_{0}$.

Newton showed that the complex roots of a polynomial (if any) appear in conjugate pairs: if $a+b i$ is a root of $p(x)$, so is $a-b i$. Descartes gave an algorithm for finding all rational roots (if any) of a polynomial $p(x)$ with integer coefficients, as follows. Let $p(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n}$. If $a / b$ is a rational root of $p(x)$, with $a$ and $b$ relatively prime, then $a$ must be a divisor of $a_{0}$ and $b$ a divisor of $a_{n}$; since $a_{0}$ and $a_{n}$ have finitely many divisors, this result determines in a finite number of steps all rational roots of $p(x)$ (note that not every $a / b$ for which $a$ divides $a_{0}$ and $b$ divides $a_{n}$ is a rational root of $\left.p(x)\right)$. He also stated (without proof) what came to be known as Descartes’ Rule of Signs: the number of positive roots of a polynomial $p(x)$ does not exceed the number of changes of sign of its coefficients (from “+” to “-” or from “-” to “+”), and the number of negative roots is at most the number of times two “+” signs or two “-” signs are found in succession.

数学代写|抽象代数作业代写abstract algebra代考|Symbolical algebra

The study of the solution of polynomial equations inevitably leads to the study of the nature and properties of various number systems, for of course the solutions are themselves numbers. Thus (as we noted) the study of number systems constitutes an important aspect of classical algebra.

The negative and complex numbers, although usêd frequently in the eighteenth century (the FTA made them inescapable), were often viewed with misgivings and were little understood. For example, Newton described negative numbers as quantities “less than nothing,” and Leibniz said that a complex number is “an amphibian between being and nonbeing.” Here is Euler on the subject: “We call those positive quantities, before which the sign $+$ is found; and those are called negative quantities, which are affected by the sign $-. “$

Although rules for the manipulation of negative numbers, such as $(-1)(-1)=1$, had been known since antiquity, no justification had in the past been given. (Euler argued that $(-a)(-b)$ must equal $a b$, for it cannot be $-a b$ since that had been “shown” to be $(-a) b$.) During the late eighteenth and early nineteenth centuries, mathematicians began to ask why such rules should hold. Members of the Analytical Society at Cambridge University made important advances on this question. Mathematics at Cambridge was part of liberal arts studies and was viewed as a paradigm of absolute truths employed for the logical training of young minds. It was therefore important, these mathematicians felt, to base algebra, and in particular the laws of operation with negative numbers, on firm foundations.

The most comprehensive work on this topic was Peacock’s Treatise of Algebra of 1830 . His main idea was to distinguish between “arithmetical algebra” and “symbolical algebra.” The former referred to operations on symbols that stood only for positive numbers and thus, in Peacock’s view, needed no justification. For example, $a-(b-c)=a-b+c$ is a law of arithmetical algebra when $b>c$ and $a>b-c$. It becomes a law of symbolical algebra if no restrictions are placed on $a, b$, and $c$. In fact, no interpretation of the symbols is called for. Thus symbolical algebra was the newly founded subject of operations with symbols that need not refer to specific objects, but that obey the laws of arithmetical algebra. This enabled Peacock to formally establish various laws of algebra. For example, $(-a)(-b)$ was shown to equal $a b$ as follows:

Since $(a-b)(c-d)=a c+b d-a d-b c(* )$ is a law of arithmetical algebra whenever $a>b$ and $c>d$, it becomes a law of symbolical algebra, which holds without restriction on $a, b, c, d$. Letting $a-0$ and $c-0$ in $( *)$ yields $(-b)(-d)-b d$.

数学代写|抽象代数作业代写abstract algebra代考|History of Group Theory

This chapter will outline the origins of the main concepts, results, and theories discussed in a first course on group theory. These include, for example, the concepts of (ahstract) group, normal subgroup, quotient group, simple group, free group, isnmorphism, homomorphism, automorphism, composition series, direct product; the theorems of Lagrange, Cauchy, Cayley, Jordan-Hölder; the theories of permutation groups and of abelian groups.

Before dealing with these issues, we wish to mention the context within mathematics as a whole, and within algebra in particular, in which group theory developed. Our “story” concerning the evolution of group theory begins in 1770 and extends to the twentieth century, but the major developments occurred in the nineteenth century. Some of the general mathematical features of that century which had a bearing on the evolution of group theory are: (a) an increased concern for rigor; (b) the emergence of abstraction; (c) the rebirth of the axiomatic method; (d) the view of mathematics as a human activity, possible without reference to, or motivation from, physical situations.
Up to about the end of the eighteenth century algebra consisted, in large part, of the study of solutions of polynomial equations. In the twentieth century algebra became the study of abstract, axiomatic systems. The transition from the so-called classical algebra of polynomial equations to the so-called modern algebra of axiomatic systems occurred in the nineteenth century. In addition to group theory, there emerged the structures of commutative rings, fields, noncommutative rings, and vector spaces. These developed alongside, and sometimes in conjunction with, group theory. Thus Galois theory involved both groups and fields; algebraic number theory contained elements of group theory in addition to commutative ring theory and field theory; group representation theory was a mix of group theory, noncommutative algebra, and linear algebra.

数学代写|抽象代数作业代写abstract algebra代考|The theory of equations and the Fundamental Theorem of Algebra

抽象代数代写

数学代写|抽象代数作业代写abstract algebra代考|The theory of equations and the Fundamental Theorem of Algebra

Viète 和 Descartes 的工作分别在 16 世纪末和 17 世纪初,将注意力的焦点从数值方程的可解性转移到了对具有字面系数的方程的理论研究。多项式方程的理论开始出现。其主要关注点是确定此类方程的根的存在性、性质和数量。具体来说:
(i) 每个多项式方程都有根吗?如果有,它是什么类型的根?这是有关该主题的所有问题中最重要和最困难的问题。事实证明,问题的第一部分比第二部分更容易回答。代数基本定理 (FTA) 回答了这两个问题:每一个具有实数或复数系数的多项式方程都有一个复数根。
(ii) 多项式方程有多少个根?笛卡尔在他的《几何学》中证明了因子定理,即如果一种是多项式的根p(X), 然后X−一种是一个因素,也就是说,p(X)=(X−一种)q(X), 在哪里q(X)是一个多项式

比的少一p(X). 重复这个过程(正式地,使用归纳法),可以得出一个多项式n正好有n根,因为它有一个根,这是由 FTA 保证的。这n根不必不同。那么,这个结果意味着,如果p(X)有学位n, 存在n数字一种1,一种2,…,一种n这样p(X)=(X−一种1)(X−一种2)…(X−一种n). FTA 保证一种一世是复数。(请注意,我们可以互换地谈论根一种多项式的p(X)和根一种多项式方程的p(X)=0; 都意味着p(一种)=02)
(iii) 我们能否确定根何时是理性的、真实的、复杂的、积极的?每个具有实系数的奇次多项式都有一个实根。这在 17 世纪和 18 世纪被直觉接受,并在 19 世纪被正式确立为微积分中中值定理的一个简单结果,它说(在此处需要的版本中)连续函数F(X)这对于某些值是积极的X对其他人来说是负数,对某些人来说必须为零X0.

牛顿证明多项式的复根(如果有)出现在共轭对中:如果一个+b一世是一个根p(X), 也是一个−b一世. 笛卡尔给出了一个算法来找到多项式的所有有理根(如果有的话)p(X)具有整数系数,如下所示。让p(X)=一个0+一个1X+⋯+一个nXn. 如果一个/b是一个有理根p(X), 和一个和b相对素数,那么一个必须是的除数一个0和b一个除数一个n; 自从一个0和一个n有有限多个除数,这个结果在有限的步数内确定了所有有理根p(X)(请注意,不是每个一个/b为此一个划分一个0和b划分一个n是一个有理根p(X)). 他还陈述了(没有证明)后来被称为笛卡尔符号规则的东西:多项式的正根数p(X)不超过其系数符号变化的次数(从“+”到“-”或从“-”到“+”),负根数最多为两个“+”号的次数或连续发现两个“-”号。

数学代写|抽象代数作业代写abstract algebra代考|Symbolical algebra

对多项式方程解的研究必然导致对各种数系的性质和性质的研究,因为解本身就是数。因此(正如我们所指出的)数系统的研究构成了经典代数的一个重要方面。

负数和复数虽然在 18 世纪被频繁使用(FTA 使它们不可避免),但人们常常怀着疑虑看待它们,而且很少被理解。例如,牛顿将负数描述为“小于无”的量,莱布尼茨说复数是“介于存在与非存在之间的两栖动物”。这是欧拉关于这个主题的:“我们称这些正量,在它们之前的符号+被发现;那些被称为负量,受符号影响−.“

尽管处理负数的规则,例如(−1)(−1)=1,自古以来就为人所知,过去没有给出任何理由。(欧拉认为(−一个)(−b)必须相等一个b, 因为它不可能−一个b因为那已经“显示”为(−一个)b.) 在 18 世纪末和 19 世纪初,数学家开始质疑为什么这样的规则应该成立。剑桥大学分析学会的成员在这个问题上取得了重要进展。剑桥的数学是文科研究的一部分,被视为一种绝对真理的范式,用于对年轻人进行逻辑训练。因此,这些数学家认为,将代数,尤其是负数运算法则建立在坚实的基础上是很重要的。

关于这个主题最全面的工作是孔雀 1830 年的《代数论文》。他的主要思想是区分“算术代数”和“符号代数”。前者指的是仅代表正数的符号的运算,因此,在孔雀看来,不需要任何理由。例如,一个−(b−C)=一个−b+C是算术代数定律,当b>C和一个>b−C. 如果没有限制,它就变成了符号代数定律一个,b, 和C. 事实上,不需要对符号进行解释。因此,符号代数是新成立的运算主题,其符号不必指代特定对象,而是遵守算术代数定律。这使孔雀能够正式建立各种代数定律。例如,(−一个)(−b)被证明等于一个b如下:

自从(一个−b)(C−d)=一个C+bd−一个d−bC(∗)是算术代数定律一个>b和C>d,它成为一个符号代数定律,它不受限制地成立一个,b,C,d. 让一个−0和C−0在(∗)产量(−b)(−d)−bd.

数学代写|抽象代数作业代写abstract algebra代考|History of Group Theory

本章将概述第一堂群论课程中讨论的主要概念、结果和理论的起源。这些包括,例如(抽象)群、正规子群、商群、单群、自由群、ismorphism、同态、自同构、复合级数、直积的概念;Lagrange、Cauchy、Cayley、Jordan-Hölder 的定理;置换群和阿贝尔群的理论。

在处理这些问题之前,我们希望提及整个数学中的背景,特别是在代数中,群论是在其中发展起来的。我们关于群论演变的“故事”始于 1770 年,一直延伸到 20 世纪,但主要的发展发生在 19 世纪。那个世纪对群论的发展产生影响的一些一般数学特征是:(a)对严谨性的日益关注;(b) 抽象的出现;(c) 公理化方法的重生;(d) 将数学视为人类活动的观点,可能无需参考或来自实际情况的动机。
直到大约 18 世纪末,代数在很大程度上还包括对多项式方程解的研究。在 20 世纪,代数成为对抽象的公理系统的研究。从所谓的经典多项式方程代数到所谓的现代公理系统代数的转变发生在 19 世纪。除了群论之外,还出现了交换环、场、非交换环和向量空间的结构。这些与群论一起发展,有时与群论一起发展。因此,伽罗瓦理论既涉及群体,也涉及领域;除了交换环论和场论之外,代数数论还包含群论的元素;群表示理论是群论、非交换代数和线性代数的混合体。

数学代写|抽象代数作业代写abstract algebra代考 请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题,以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法,其中不假设数据来自于由少数参数决定的规定模型;这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型(GLM)归属统计学领域,是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。

有限元方法代写

有限元方法(FEM)是一种流行的方法,用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

tatistics-lab作为专业的留学生服务机构,多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务,包括但不限于Essay代写,Assignment代写,Dissertation代写,Report代写,小组作业代写,Proposal代写,Paper代写,Presentation代写,计算机作业代写,论文修改和润色,网课代做,exam代考等等。写作范围涵盖高中,本科,研究生等海外留学全阶段,辐射金融,经济学,会计学,审计学,管理学等全球99%专业科目。写作团队既有专业英语母语作者,也有海外名校硕博留学生,每位写作老师都拥有过硬的语言能力,专业的学科背景和学术写作经验。我们承诺100%原创,100%专业,100%准时,100%满意。

随机分析代写


随机微积分是数学的一个分支,对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。

回归分析代写

多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。

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。

R语言代写问卷设计与分析代写
PYTHON代写回归分析与线性模型代写
MATLAB代写方差分析与试验设计代写
STATA代写机器学习/统计学习代写
SPSS代写计量经济学代写
EVIEWS代写时间序列分析代写
EXCEL代写深度学习代写
SQL代写各种数据建模与可视化代写

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