### 数学代写|抽象代数作业代写abstract algebra代考|History of Classical Algebra

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## 数学代写|抽象代数作业代写abstract algebra代考|Early roots

For about three millennia, until the early nineteenth century, “algebra” meant solving polynomial equations, mainly of degree four or less. Questions of notation for such equations, the nature of their roots, and the laws governing the various number systems to which the roots belonged, were also of concern in this connection. All these matters became known as classical algebra. (The term “algebra” was only coined in the ninth century AD.) By the early decades of the twentieth century, algebra had evolved into the study of axiomatic systems. The axiomatic approach soon came to be called modern or abstract algebra. The transition from classical to modern algebra occurred in the nineteenth century.

Most of the major ancient civilizations, the Babylonian, Egyptian, Chinese, and $\mathrm{~ H i n d u , ~ d e ́ a ̂ l t ~ w i t h ~ t h e ~ s o l u t i o n ~ o f ~ p o o l y n o o m i a ̉ l ~ e ̨ q u a t i o n s , ~ m a ̂ i n l y ~ l i n e ́ a ̂ ́ ~ a}$ equations. The Babylonians (c. $1700 \mathbf{B C}$ ) were particularly proficient “algebraists.” They were able to solve quadratic equations, as well as equations that lead to quadratic equations, for example $x+y=a$ and $x^{2}+y^{2}=b$, by methods similar to ours. The equations were given in the form of “word problems.” Here is a typical example and its solution:
I have added the area and two-thirds of the side of my square and it is $0 ; 35$ [35/60 in sexagesimal notation]. What is the side of my square?
In modern notation the problem is to solve the equation $x^{2}+(2 / 3) x=35 / 60$. The solution given by the Babylonians is:
You take 1, the coefficient. Two-thirds of 1 is $0 ; 40$. Half of this, $0 ; 20$, you multiply by $0 ; 20$ and it [the result] $0 ; 6,40$ you add to $0 ; 35$ and [the result] $0 ; 41,40$ has $0 ; 50$ as its square root. The $0 ; 20$, which you have multiplied by itself, you subtract from $0 ; 50$, and $0 ; 30$ is [the side of] the square.
The instructions for finding the solution can be expressed in modern notation as $x=\sqrt{[(0 ; 40) / 2]^{2}+0 ; 35}-(0 ; 40) / 2=\sqrt{0 ; 6,40+0 ; 35}-$ $0 ; 20=\sqrt{0 ; 41,40}-0 ; 20=0 ; 50-0 ; 20=0 ; 30$.

## 数学代写|抽象代数作业代写abstract algebra代考|The Greeks

The mathematics of the ancient Greeks, in particular their geometry and number theory, was relatively advanced and sophisticated, but their algebra was weak. Euclid’s great work Elements (c. $300 \mathrm{BC}$ ) contains several parts that have been interpreted by historians, with notable exceptions (e.g., $[14,16]$ ), as algebraic. These are geometric propositions that, if translated into algebraic language, yield algebraic results: laws of algebra as well as solutions of quadratic equations. This work is known as geometric algebra.

For example, Proposition II.4 in the Elements states that “If a straight line be cut at random, the square on the whole is equal to the square on the two parts and twice the rectangle contained by the parts.” If $a$ and $b$ denote the parts into which the straight line is cut, the proposition can be stated algebraically as $(a+b)^{2}=a^{2}+2 a b+b^{2}$.

Proposition II.11 states: “To cut a given straight line so that the rectangle contained by the whole and one of the segments is equal to the square on the remaining segment.” It asks, in algebraic language, to solve the equation $a(a-x)=x^{2}$. See [7, p. 70].
Note that Greek algebra, such as it is, speaks of quantities rather than numbers. Moreover, homogeneity in algebraic expressions is a strict requirement; that is, all terms in such expressions must be of the same degree. For example, $x^{2}+x=b^{2}$ would not be admitted as a legitimate equation. See [1], [2], [18], [19].

A much more significant Greek algebraic work is Diophantus’ Arithmetica (c. $250 \mathrm{AD}$ ). Although essentially a book on number theory, it contains solutions of equations in integers or rational numbers. More importantly for progress in algebra, it introduced a partial algebraic notation-a most important achievement: $\varsigma$ denoted an unknown, $\Phi$ negation, i $\sigma$ equality, $\Delta^{\sigma}$ the square of the unknown, $K^{\sigma}$ its cube, and $M$ the absence of the unknown (what we would write as $x^{0}$ ). For example, $x^{3}-2 x^{2}+10 x-1=5$ would be written as $K^{\sigma} \alpha \zeta$ í $\Phi \Delta^{\sigma} \beta M \alpha$ í $\sigma M \varepsilon$ (numbers were denoted by letters, so that, for example, $\alpha$ stood for 1 and $\varepsilon$ for 5 ; moreover, there was no notation for addition, thus all terms with positive coefficients were written first, followed by those with negative coefficients).
(a) He gave two basic rules for working with algebraic expressions: the transfer of a term from one side of an equation to the other, and the elimination of like terms from the two sides of an equation.
(b) He defined negative powers of an unknown and enunciated the law of exponents, $x^{m} x^{n}=x^{m+n}$, for $-6 \leq m, n, m+n \leq 6 .$
(c) He stated several rules for operating with negative coefficients, for example: “deficiency multiplied by deficiency yields availability” $((-a)(-b)=a b)$.
(d) He did away with such staples of the classical Greek tradition as (i) giving a geometric interpretation of algebraic expressions, (ii) restricting the product of terms to degree at most three, and (iii) requiring homogeneity in the terms of an algebraic expression. See [1], [7], [18].

## 数学代写|抽象代数作业代写abstract algebra代考|Al-Khwarizmi

Islamic mathematicians attained important algebraic accomplishments between the ninth and fifteenth centuries AD. Perhaps foremost among them was Muhammad ibnMusa al-Khwarizmi (c. 780-850), dubbed by some “the Euclid of algebra” because hé systēmaatized the subject (ās it then existēd) and made it into an independent fièld of study. He did this in his hook al-jahr w al-muqahalah. “Al-jahr” (from which stems our word “algebra”) denotes the moving of a negative term of an equation to the other side so as to make it positive, and “al-muqabalah” refers to cancelling equal (positive) terms on both sides of an equation. These are, of course, basic procedures for solving polynomial equations. Al-Khwarizmi (from whose name the term “algorithm” is derived) applied them to the solution of quadratic equations. He classified these into five types: $a x^{2}=b x, a x^{2}=b, a x^{2}+b x=c, a x^{2}+c=b x$, and $a x^{2}=b x+c$. This categorization was necessary since al-Khwarizmi did not admit negative coefficients or zero. He also had essentially no notation, so that his problems and solutions were expressed rhetorically. For example, the first and third equations above were given as: “squares equal roots” and “squares and roots equal numbers” (an unknown was called a “root”). Al-Khwarizmi did offer justification, albeit geometric, for his solution procedures. See [13], [17].

## 数学代写|抽象代数作业代写abstract algebra代考|The Greeks

(a) 他给出了处理代数表达式的两个基本规则：将一项从方程的一侧转移到另一侧，以及从方程的两侧消除相似的项。一个方程。
(b) 他定义了未知数的负幂并阐明了指数定律，X米Xn=X米+n， 为了−6≤米,n,米+n≤6.
(c) 他陈述了使用负系数操作的几条规则，例如：“缺陷乘以缺陷产生可用性”((−一个)(−b)=一个b).
(d) 他摒弃了古典希腊传统中的一些主要内容：(i) 给出代数表达式的几何解释，(ii) 将项的乘积限制为最多三阶，以及 (iii) 要求项的同质性一个代数表达式。见 [1]、[7]、[18]。

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