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# 数学代写|现代代数代考Modern Algebra代写|MAT423/523 Fields

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## 数学代写|现代代数代考Modern Algebra代写|Fields

Informally, a field is a set equipped with four operations – addition, subtraction, multiplication, and division that have the usual properties. (They don’t have to have the other operations that $\mathbf{R}$ has, like powers, roots, logs, and the myriad other functions like $\sin x$.)
Definition $1.1$ (Field). A field is a set equipped with two binary operations, one called addition and the other called multiplication, denoted in the usual manner, which are both commutative and associative, both have identity elements (the additive identity denoted 0 and the multiplicative identity denoted 1), addition has inverse elements (the inverse of $x$ denoted $-x$ ), multiplication has inverses of nonzero elements (the inverse of $x$ denoted $\frac{1}{x}$ or $\left.x^{-1}\right)$, multiplication distributes over addition, and $0 \neq 1$.
This definition will be spelled out in detail in chapter 2.
Of course, one example of a field in the field of real numbers $\mathbf{R}$. What are some others?
Example 1.2 (The field of rational numbers, Q). Another example is the field of rational numbers. A rational number is the quotient of two integers $a / b$ where the denominator is not 0 . The set of all rational numbers is denoted $\mathbf{Q}$. We’re familiar with the fact that the sum, difference, product, and quotient (when the denominator is not zero) of rational numbers is another rational number, so $\mathbf{Q}$ has all the operations it needs to be a field, and since it’s part of the field of the real numbers $\mathbf{R}$, its operations have the the properties necessary to be a field. We say that $\mathbf{Q}$ is a subfield of $\mathbf{R}$ and that $\mathbf{R}$ is an extension of $\mathbf{Q}$. But $\mathbf{Q}$ is not all of $\mathbf{R}$ since there are irrational numbers like $\sqrt{2}$.

Example 1.3 (The field of complex numbers, C). Yet another example is the field of complex numbers C. A complex number is a number of the form $a+b i$ where $a$ and $b$ are real numbers and $i^2=-1$. The field of real numbers $\mathbf{R}$ is a subfield of $\mathbf{C}$. We’ll review complex numbers before we use them. See Dave’s Short Course on Complex Numbers at http://www . clarku. edu/ djoyce/complex

## 数学代写|现代代数代考Modern Algebra代写|Rings

Rings will have the three operations of addition, subtraction, and multiplication, but don’t necessarily have division. Most of our rings will have commutative multiplication, but some won’t, so we won’t require that multiplication be commutative in our definition. All the rings we’ll look at have a multiplicative identity, 1, so we’ll include that in the definition.

Definition $1.4$ (Ring). A ring is a set equipped with two binary operations, one called addition and the other called multiplication, denoted in the usual manner, which are both associative, addition is commutative, both have identity elements (the additive identity denoted 0 and the multiplicative identity denoted 1), addition has inverse elements (the inverse of $x$ denoted $-x$ ), and multiplication distributes over addition. If multiplication is also commutative, then the ring is called a commutative ring.

Of course, all fields are automatically rings, in fact commutative rings, but what are some other rings?

Example 1.5 (The ring of integers, $\mathbf{Z}$ ). The ring of integers $\mathbf{Z}$ includes all integers (whole numbers)-positive, negative, or 0 . Addition, subtraction, and multiplication satisfy the requirements for a ring, indeed, a commutative ring. But there are no multiplicative inverses for any elements except 1 and $-1$. For instance, $1 / 2$ is not an integer. We’ll find that although the ring of integers looks like it has less structure than a field, this very lack of structure allows us to discover more about integers. We’ll be able to talk about prime numbers, for example.
Example 1.6 (Polynomial rings). A whole family of examples are the rings of polynomials. Let $R$ be any commutative ring (perhaps a field), and let $R[x]$ include all polynomials with coefficients in $R$. We know how to add, subtract, and multiply polynomials, and these operations have the properties required to make $R[x]$ a commutative ring. We have, for instance, the ring of polynomials with real coefficients $\mathbf{R}[x]$, the ring with integral coefficients $\mathbf{Z}[x]$, etc.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|Rings

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## 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。