### 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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## 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

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

(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.

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