## 数学代写|伽罗瓦理论代写Galois Theory代考|МАТН3040

statistics-lab™ 为您的留学生涯保驾护航 在代写伽罗瓦理论Galois Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写伽罗瓦理论Galois Theory代写方面经验极为丰富，各种代写伽罗瓦理论Galois Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|伽罗瓦理论代写Galois Theory代考|Quintic Equations

So far, we have a series of special tricks, different in each case. We can approach the general quintic equation
$$t^5+a t^4+b t^3+c t^2+d t+e=0$$
in a similar way. A Tschirnhaus transformation $y=t+a / 5$ reduces it to
$$y^5+p y^3+q y^2+r y+s=0$$
However, all variations on the tricks that we used for the quadratic, cubic, and quartic equations grind to a halt.

In 1770-1771 Lagrange analysed all of the above special tricks, showing that they can all be ‘explained’ using general principles about symmetric functions of the roots. When he applied this method to the quintic, however, he found that it ‘reduced’ the problem to a sextic – an equation of degree 6. Instead of helping, the method made the problem worse.

Lagrange observed that all methods for solving polynomial equations by radicals involve constructing rational functions of the roots that take a small number of values when the roots $\alpha_j$ are permuted. Prominent among these is the expression
$$\delta=\prod_{1 \leq j<k \leq n}\left(\alpha_j-\alpha_k\right)$$
where $n$ is the degree. This takes just two values, $\pm \delta$ : plus for even permutations and minus for odd ones. Therefore $\Delta=\delta^2$ (known as the discriminant because it is nonzero precisely when the roots are distinct, so it ‘discriminates’ among the roots) is a rational function of the coefficients. This gets us started, and it yields a complete solution for the quadratic, but for cubics upwards it does not help much unless we can find other expressions in the roots with similar properties under permutation.

Lagrange worked out what these expressions look like for the cubic and the quartic, and noticed a pattern. For example, if a cubic polynomial has roots $\alpha_1, \alpha_2, \alpha_3$, and $\omega$ is a primitive cube root of unity, then the expression
$$u=\left(\alpha_1+\omega \alpha_2+\omega^2 \alpha_3\right)^3$$ takes exactly two distinct values.

## 数学代写|伽罗瓦理论代写Galois Theory代考|The Fundamental Theorem of Algebra

At the time of Galois, the natural setting for most mathematical investigations was the complex number system. The real numbers were inadequate for many questions, because $-1$ has no real square root. The arithmetic, algebra, anddecisively – analysis of complex numbers were richer, more elegant, and more complete than the corresponding theories for real numbers.

In this chapter we establish one of the key properties of $\mathbb{C}$, known as the Fundamental Theorem of Algebra. This theorem asserts that every polynomial equation with coefficients in $\mathbb{C}$ has a solution in $\mathbb{C}$. This theorem is, of course, false over $\mathbb{R}$-consider the equation $t^2+1=0$. It was fundamental to classical algebra, but the name is somewhat archaic, and modern algebra bypasses $\mathbb{C}$ altogether, preferring greater generality. Because we find it convenient to work in the same setting as Galois, the theorem is fundamental for us.

All rigorous proofs of the Fundamental Theorem of Algebra require quite a lot of background. Here, we give a proof that uses a few simple ideas from algebra and trigonometry, estimates of the kind that are familiar from any first course in analysis, and one simple basic result from point-set topology. Later, we give an almost purely algebraic proof, but the price is the need for much more machinery: see Chapter 23. Ironically, that proof uses Galois theory to prove the Fundamental Theorem of Algebra, the exact opposite of what Galois did. The logic is not circular, because the proof in Chapter 23 rests on the abstract approach to Galois theory described in the second part of this book, which makes no use of the Fundamental Theorem of Algebra.

# 伽罗瓦理论代考

## 数学代写|伽罗瓦理论代写Galois Theory代考|Quintic Equations

$$t^5+a t^4+b t^3+c t^2+d t+e=0$$

$$y^5+p y^3+q y^2+r y+s=0$$

$$\delta=\prod_{1 \leq j<k \leq n}\left(\alpha_j-\alpha_k\right)$$

$$u=\left(\alpha_1+\omega \alpha_2+\omega^2 \alpha_3\right)^3$$

## 有限元方法代写

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。

## 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。

## 数学代写|伽罗瓦理论代写Galois Theory代考|МАТH0701

statistics-lab™ 为您的留学生涯保驾护航 在代写伽罗瓦理论Galois Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写伽罗瓦理论Galois Theory代写方面经验极为丰富，各种代写伽罗瓦理论Galois Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|伽罗瓦理论代写Galois Theory代考|Solving Equations

We tend to think of the great problems of mathematics as being things like Fermat’s Last Theorem, the Poincaré Conjecture, or the Riemann Hypothesis: problems of central importance that remained unsolved for decades or even centuries. But the really big problems of mathematics are more general. A problem that runs like an ancient river through the middle of the territory we are going to explore is: Find out how to solve equations. Or, as often as not, prove that it cannot be done with specified methods. What sort of equations? There are many kinds: Diophantine equations, differential equations (ordinary, partial, or delay), difference equations, integral equations, operator equations … For Galois, it was polynomial equations. We work up to those in easy stages.

Historically, new kinds of number like $\sqrt{2}$ or i were introduced because the old ones were inadequate for solving some important problems. Most such problems can be formulated using equations, though it must be said that this is a modern interpretation, and the ancient mathematicians did not think in quite those terms.

For example, the step from $\mathbb{N}$ to $\mathbb{Z}$ is needed because although some equations, such as
$$t+2=7$$
can be solved for $t \in \mathbb{N}$, others, such as
$$t+7=2$$
cannot. However, such equations can be solved in $\mathbb{Z}$, where $t=-5$ makes sense. (The symbol $x$ is more traditional than $t$ here, but it is convenient to standardise on $t$ for the rest of the book, so we may as well start straight away.)

Similarly, the step from $\mathbb{Z}$ to $\mathbb{Q}$ (historically, it was initially from $\mathbb{N}$ to $\mathbb{Q}^{+}$, the positive rationals) makes it possible to solve the equation
$$2 t=7$$
because $t=\frac{7}{2}$ makes sense in $\mathbb{Q}$.

## 数学代写|伽罗瓦理论代写Galois Theory代考|Peculiarities of Cardano’s Formula

An old warning, which goes back to Aesop’s Fables, is: ‘Be careful what you wish for: you might get it’. We have wished for a formula for the solution, and we have got one. It has its peculiarities.

First: recall that over $\mathbb{C}$, every nonzero complex number $z$ has three cube roots. If one of them is $\alpha$, then the other two are $\omega \alpha$ and $\omega^2 \alpha$, where
$$\omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2}$$
is a primitive cube root of 1 . Then
$$\omega^2=-\frac{1}{2}-\mathrm{i} \frac{\sqrt{3}}{2}$$
The expression for $y$ therefore appears to lead to nine solutions of the form
$$\begin{array}{lcr} \alpha+\beta & \alpha+\omega \beta & \alpha+\omega^2 \beta \ \omega \alpha+\beta & \omega \alpha+\omega \beta & \omega \alpha+\omega^2 \beta \ \omega^2 \alpha+\beta & \omega^2 \alpha+\omega \beta & \omega^2 \alpha+\omega^2 \beta \end{array}$$
where $\alpha, \beta$ are specific choices of the cube roots.
However, not all of these expressions are zeros. Equation (1.5) implies (1.7), but (1.7) implies (1.5) only when we make the correct choices of cube roots. If we choose $\alpha, \beta$ so that $3 \alpha \beta+p=0$, then the solutions are
$$\alpha+\beta \quad \omega \alpha+\omega^2 \beta \quad \omega^2 \alpha+\omega \beta$$
Another peculiarity emerges when we try to solve equations whose solutions we already know. For example,
$$y^3+3 y-36=0$$

# 伽罗瓦理论代考

## 数学代写|伽罗瓦理论代写Galois Theory代考|Solving Equations

$$t+2=7$$

$$t+7=2$$

$$2 t=7$$

## 数学代写|伽罗瓦理论代写Galois Theory代考|Peculiarities of Cardano’s Formula

$$\omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2}$$

$$\omega^2=-\frac{1}{2}-\mathrm{i} \frac{\sqrt{3}}{2}$$

$$\alpha+\beta \quad \alpha+\omega \beta \quad \alpha+\omega^2 \beta \omega \alpha+\beta \quad \omega \alpha+\omega \beta \quad \omega \alpha+\omega^2 \beta \omega^2 \alpha+\beta \quad \omega^2 \alpha+\omega \beta \quad \omega^2 \alpha+\omega^2 \beta$$

$$\alpha+\beta \quad \omega \alpha+\omega^2 \beta \quad \omega^2 \alpha+\omega \beta$$

$$y^3+3 y-36=0$$

## 有限元方法代写

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。

## 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。

## 数学代写|伽罗瓦理论代写Galois Theory代考|MX4082

statistics-lab™ 为您的留学生涯保驾护航 在代写伽罗瓦理论Galois Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写伽罗瓦理论Galois Theory代写方面经验极为丰富，各种代写伽罗瓦理论Galois Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|伽罗瓦理论代写Galois Theory代考|Complex Numbers

A complex number has the form
$$z=x+\mathrm{i} y$$
where $x, y$ are real numbers and $\mathrm{i}^2=-1$. Therefore $\mathrm{i}=\sqrt{-1}$, in some sense.
Throughout, we use the Roman letter $\mathrm{i}$ for $\sqrt{-1}$. This frees up italic $i$ for other uses.

The easiest way to define what we mean by $\sqrt{-1}$ is to consider $\mathbb{C}$ as the set $\mathbb{R}^2$ of all pairs of real numbers $(x, y)$, with algebraic operations
\begin{aligned} \left(x_1, y_1\right)+\left(x_2, y_2\right) &=\left(x_1+x_2, y_1+y_2\right) \ \left(x_1, y_1\right)\left(x_2, y_2\right) &=\left(x_1 x_2-y_1 y_2, x_1 y_2+x_2 y_1\right) \end{aligned}
Then we identify $(x, 0)$ with the real number $x$ to arrange that $\mathbb{R} \subseteq \mathbb{C}$, and define $\mathrm{i}=(0,1)$. In consequence, $(x, y)$ becomes identified with $x+\mathrm{i} y$. The formulas (1.1) imply that $\mathrm{i}^2=(0,1)(0,1)=(-1,0)$, which is identified with the real number $-1$, so $\mathrm{i}$ is a ‘square root of minus one’. Observe that $(0,1)$ is not of the form $(x, 0)$, so $\mathrm{i}$ is not real, which is as it should be, since $-1$ has no real square root.

This approach seems first to have been published by the Irish mathematician William Rowan Hamilton in 1837, but in that year, Gauss wrote to the geometer Wolfgang Bolyai that the same idea had occurred to him in 1831. This was probably true, because Gauss usually worked things out before anybody else did, but he set himself such high standards for publication that many of his more important ideas never saw print under his name. Moreover, Gauss was somewhat conservative and shied away from anything potentially controversial.

Once we see that complex numbers are just pairs of real numbers, the previously mysterious status of the ‘imaginary’ number $\sqrt{-1}$ becomes much more prosaic. In fact, to the modern eye, it is the ‘real’ numbers that are mysterious, because their rigorous definition involves analytic ideas such as sequences and convergence, which lead into deep philosophical waters and axiomatic set theory. In contrast, the step from $\mathbb{R}$ to $\mathbb{R}^2$ is essentially trivialexcept for the peculiarities of human psychology.

## 数学代写|伽罗瓦理论代写Galois Theory代考|Subfields and Subrings of the Complex Numbers

Abstract algebra courses usually introduce (at least) three basic types of algebraic structure, defined by systems of axioms: groups, rings, and fields. Linear algebra adds a fourth, vector spaces, which are so important that they usually warrant a separate lecture course. For the first half of this book, we steer clear of abstract rings and fields, but we do assume the basics of finite group theory and linear algebra.

Recall that a group is a set $G$ equipped with an operation of ‘multiplication’ written $(g, h) \mapsto g h$. If $g, h \in G$, then $g h \in G$. The associative law $(g h) k=g(h k)$ holds for all $g, h, k \in G$. There is an identity $1 \in G$ such that $1 g=g=g 1$ for all $g \in G$. Finally, every $g \in G$ has an inverse $g^{-1} \in G$ such that $g g^{-1}=1=g^{-1} g$. The classic example here is the symmetric group $\mathbb{S}_n$, consisting of all permutations of the set ${1,2, \ldots, n}$ under the operation of composition. We assume familiarity with these axioms, and with subgroups, isomorphisms, homomorphisms, normal subgroups, and quotient groups; see Humphreys (1996), Neumann, Stoy and Thompson (1994), or any other introductory group theory text.

Rings are sets equipped with operations of addition, subtraction, and multiplication; fields also have a notion of division. The formal definitions were supplied by Heinrich Weber in 1893 . The axioms specify the formal properties assumed for these operations – for example, the commutative law $a b=b a$ for multiplication.

In the first part of this book, we do not assume familiarity with abstract rings and fields. Instead, we restrict attention to subrings and subfields of $\mathbb{C}$, or polynomials and rational functions over such subrings and subfields. Informally, we assume that the terms ‘polynomial’ and ‘rational expression’ (or ‘rational function’) are familiar, at least over $\mathbb{C}$, although for safety’s sake we define them when the discussion becomes more formal, and redefine them when we make the whole theory more abstract in the second part of the book. There were no formal concepts of ‘ring’ or ‘field’ in Galois’s day, and linear algebra was in a rudimentary state. He had to invent groups for himself. So we are still permitting ourselves a more extensive conceptual toolkit than his.

# 伽罗瓦理论代考

## 数学代写|伽罗瓦理论代写Galois Theory代考|Complex Numbers

$$z=x+\mathrm{i} y$$

$$\left(x_1, y_1\right)+\left(x_2, y_2\right)=\left(x_1+x_2, y_1+y_2\right)\left(x_1, y_1\right)\left(x_2, y_2\right) \quad=\left(x_1 x_2-y_1 y_2, x_1 y_2+x_2 y_1\right)$$

## 有限元方法代写

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。

## 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。