### 数学代写|抽象代数作业代写abstract algebra代考|Cubic and quartic equations

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## 数学代写|抽象代数作业代写abstract algebra代考|Cubic and quartic equations

The Babylonians were solving quadratic equations by about $1600 \mathrm{BC}$, using essentially the equivalent of the quadratic formula. A natural question is therefore whether cubic equations could be solved using similar formulas (see below). Another three thousand years would pass before the answer would be known. It was a great event

in algebra when mathematicians of the sixteenth century succeeded in solving “by radicals” not only cubic but also quartic equations.

A solution by radicals of a polynomial equation is a formula giving the roots of the equation in terms of its coefficients. The only permissible operations to be applied to the coefficients are the four algebraic operations (addition, subtraction, multiplication, and division) and the extraction of roots (square roots, cube roots, and so on, that is, “radicals”). For example, the quadratic formula $x=\left(-b \pm \sqrt{b^{2}-4 a c}\right) / 2 a$ is a solution by radicals of the equation $a x^{2}+b x+c=0$.

A solution by radicals of the cubic was first published by Cardano in The Great Art (referring to algebra) of 1545 , but it was discovered earlier by del Ferro and by Tartaglia. The latter had passed on his method to Cardano, who had promised that he would not publish it, which he promptly did. What came to be known as Cardano’s formula for the solution of the cubic $x^{3}=a x+b$ was given by
$$x=\sqrt[3]{b / 2+\sqrt{(b / 2)^{2}-(a / 3)^{3}}}+\sqrt[3]{b / 2-\sqrt{(b / 2)^{2}-(a / 3)^{3}}}$$

## 数学代写|抽象代数作业代写abstract algebra代考|The cubic and complex numbers

Mathematicians adhered for centuries to the following view with respect to the square roots of negative numbers: since the squares of positive as well as of negative numbers are positive, square roots of negative numbers do not-in fact, cannot-exist. All this changed following the solution by radicals of the cubic in the sixteenth century.
Square roots of negative numbers arise “naturally” when Cardano’s formula (see p. 6) is used to solve cubic equations. For example, application of his formula to the equation $x^{3}=9 x+2$ gives $x=\sqrt[3]{2 / 2+\sqrt{(2 / 2)^{2}-(9 / 3)^{3}}}+$ $\sqrt[3]{2 / 2-\sqrt{(2 / 2)^{2}-(9 / 3)^{3}}}=\sqrt[3]{1+\sqrt{-26}}+\sqrt[3]{1-\sqrt{-26}}$. What is one to make of this solution? Since Cardano was suspicious of negative numbers, he certainly had no taste for their square roots, so he regarded his formula as inapplicable to equations such as $x^{3}=9 x+2$. Judged by past experience, this was not an unreasonable attitude. For example, to the Pythagoreans, the side of a square of area 2 was nonexistent (in today’s language, we would say that the equation $x^{2}=2$ is unsolvable).

All this was changed by Bombelli. In his important book Algebra (1572) he applied Cardano’s formula to the equation $x^{3}=15 x+4$ and obtained $x=\sqrt[3]{2+\sqrt{-121}}+$ $\sqrt[3]{2-\sqrt{-121}}$. But he could not dismiss the solution, for he noted by inspection that $x=4$ is a root of this equation. Moreover, its other two roots, $-2 \pm \sqrt{3}$, are also real numbers. Here was a paradox: while all three roots of the cubic $x^{3}=$ $15 x+4$ are real, the formula used to obtain them involved square roots of negative numbers-meaningless at the time. How was one to resolve the paradox?

Bombelli adopted the rules for real quantities to manipulate “meaningless” expressions of the form $a+\sqrt{-b}(b>0)$ and thus managed to show that $\sqrt[3]{2+\sqrt{-121}}=2+\sqrt{-1}$ and $\sqrt[3]{2-\sqrt{-121}}=2-\sqrt{-1}$, and hence that $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}=(2+\sqrt{-1})+(2-\sqrt{-1})=4$. Bombëlli had given meaning to the “meaningless” by thinking the “unthinkable,” namely that square roots of negative numbers could be manipulated in a meaningful way to yield

significant results. This was a very bold move on his part. As he put it:
It was a wild thought in the judgment of many; and I too was for a long time of the same opinion. The whole matter seemed to rest on sophistry rather than on truth. Yet I sought so long until I actually proved this to be the case [11].
Bombelli developed a “calculus” for complex numbers, stating such rules as $(+\sqrt{-1})(+\sqrt{-1})=-1$ and $(+\sqrt{-1})(-\sqrt{-1})=1$, and defined addition and multiplication of specific complex numbers. This was the birth of complex numbers. But birth did not entail legitimacy. For the next two centuries complex numbers were shrouded in mystery, little understood, and often ignored. Only following their geometric representation in 1831 by Gauss as points in the plane were they accepted as bona fide elements of the number system. (The earlier works of Argand and Wessel on this topic were not well known among mathematicians.) See [1], [7], [13].

Note that the equation $x^{3}=15 x+4$ is an example of an “irreducible cubic,” namely one with rational coefficients, irreducible over the rationals, all of whose roots are real. It was shown in the nineteenth century that any solution by radicals of such a cubic (not just Cardano’s) must involve complex numbers. Thus complex numbers are unavoidable when it comes to finding solutions by radicals of the irreducible cubic. It is for this reason that they arose in connection with the solution of cubic rather than quadratic equations, as is often wrongly assumed. (The nonexistence of a solution of the quadratic $x^{2}+1=0$ was readily accepted for centuries.)

## 数学代写|抽象代数作业代写abstract algebra代考|Algebraic notation: Viète and Descartes

Mathematical notation is now taken for granted. In fact, mathematics without a welldeveloped symbolic notation would be inconceivable. It should be noted, however, that the subject evolved for about three millennia with hardly any symbols. The introduction and perfection of symbolic notation in algebra occurred for the most part in the sixteenth and early seventeenth centuries, and is due mainly to Viète and Descartes.

The decisive step was taken by Viète in his Introduction to the Analytic Art (1591). He wanted to breathe new life into the method of analysis of the Greeks, a method of discovery used to solve problems, to be contrasted with their method of synthesis, used to prove theorems. The former method he identified with algebra. He saw it as “the science of correct discovery in mathematics,” and had the grand vision that it would “leave no problem unsolved.”

Viète’s basic idea was to introduce arbitrary parameters into an equation and to distinguish these from the equation’s variables. He used consonants $(B, C, D, \ldots)$ to denote parameters and vowels $(A, E, I, \ldots)$ to denote variables. Thus a quadratic equation was written as $B A^{2}+C A+D=0$ (although this was not exactly Viète’s notaation’, seé bẽlow). Tó us this āppeears to bé a simplé and natüral ideá, but it was a fundamental departure in algebra: for the first time in over three millennia one could speak of a general quadratic equation, that is, an equation with (arbitrary) literal coefficients rather than one with (specific) numerical coefficients.

## 数学代写|抽象代数作业代写abstract algebra代考|Cubic and quartic equations

Cardano 于 1545 年在 The Great Art（指代数）中首次发表了由三次根式的解法，但较早时由 del Ferro 和 Tartaglia 发现。后者将他的方法传给了卡尔达诺，卡尔达诺承诺他不会发表，他很快就做到了。什么后来被称为卡尔达诺解立方的公式X3=一个X+b由

X=b/2+(b/2)2−(一个/3)33+b/2−(b/2)2−(一个/3)33

## 数学代写|抽象代数作业代写abstract algebra代考|The cubic and complex numbers

Bombelli 采用实数规则来操纵形式的“无意义”表达式一个+−b(b>0)并因此设法证明2+−1213=2+−1和2−−1213=2−−1，因此X=2+−1213+2−−1213=(2+−1)+(2−−1)=4. Bombëlli 通过思考“不可想象”赋予“无意义”以意义，即负数的平方根可以以有意义的方式被操纵以产生

Bombelli 为复数开发了一种“微积分”，阐述了如下规则(+−1)(+−1)=−1和(+−1)(−−1)=1, 并定义了特定复数的加法和乘法。这就是复数的诞生。但出生并不意味着合法性。在接下来的两个世纪里，复数笼罩在神秘之中，鲜为人知，而且经常被忽视。只有在 1831 年高斯将它们的几何表示为平面中的点之后，它们才被接受为数系的真正元素。（Argand 和 Wessel 在这个主题上的早期著作在数学家中并不为人所知。）参见 [1]、[7]、[13]。

## 数学代写|抽象代数作业代写abstract algebra代考|Algebraic notation: Viète and Descartes

Viète 在他的《分析艺术导论》（1591 年）中迈出了决定性的一步。他想为希腊人的分析方法注入新的活力，这是一种用来解决问题的发现方法，与他们用来证明定理的综合方法形成对比。他用代数确定了前一种方法。他将其视为“数学中正确发现的科学”，并拥有“没有任何问题未解决”的宏伟愿景。

Viète 的基本思想是将任意参数引入方程，并将这些参数与方程的变量区分开来。他用辅音(乙,C,D,…)表示参数和元音(一种,和,一世,…)来表示变量。因此，一个二次方程写为乙一种2+C一种+D=0（尽管这并不完全是 Viète 的符号，参见 bẽlow）。对我们来说，这似乎是一个简单和自然的想法，但它是代数的一个根本性转变：三千年以来，人们第一次可以谈论一般的二次方程，即具有（任意）字面系数的方程，而不是比具有（特定）数字系数的。

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