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# 数学代写|现代代数代考Modern Algebra代写|MATH402 The field of rational numbers, fields of fractions

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## 数学代写|现代代数代考Modern Algebra代写|The field of rational numbers, fields of fractions

Suppose that we already have constructed the integral domain of integers $\mathbf{Z}$, but for some reason do not have the field of rational numbers $\mathbf{Q}$. Then we could construct $\mathbf{Q}$ from $\mathbf{Z}$ since each rational number can be named by a pair of integers. We’ll do that. The steps we use only depend on $\mathbf{Z}$ being an integral domain. That means that the construction we use can also be used to create a field of fractions $F$ from any integral domain $R$. In the following, think of the integral domain $R$ as bing $\mathbf{Z}$ and the field $F$ as being $\mathbf{Q}$.

An equivalence relation on pairs of integers. First of all, a rational number $\frac{m}{n}$ can be named by a pair of integers $(m, n)$ where the second integer $n$ does not equal 0 . But different pairs $(m, n)$ and $(k, l)$ can name the same integer $\frac{m}{n}=\frac{k}{l}$ if $m l=n k$. That suggests if we want to create rational numbers from integers, we’ll need an equivalence relation on pairs of elements of the integral domain $R$.

We’ll start with the set $R \times R_{\neq 0}$ of ordered pairs $(m, n)$ of elements of an integral domain $R$ with $n \neq 0$. Define a relation $\equiv$ on this set by
$$(m, n) \equiv(k, l) \quad \text { iff } \quad m l=n k$$

## 数学代写|现代代数代考Modern Algebra代写|The formal definition of categories

Unlike fields, rings, and groups, we won’t require that categories build on sets. In a category the collection of all its objects won’t be a set because the collection is larger than any set. That’s not a problem since theories don’t have to be built on set theory. Indeed, set theory itself is not built on set theory.
Definition 3.17. A category $\mathcal{C}$ consists of

objects often denoted with uppercase letters, and

morphisms (also called maps or arrows) often denoted with lowercase letters.

Each morphism $f$ has a domain which is an object and a codomain which is also an object. If the domain of $f$ is $A$ and the codomain is $B$, then we write $f: A \rightarrow B$ or $A \stackrel{f}{\rightarrow} B$. The collection of all morphisms from $A$ to $B$ is $\operatorname{denoted} \operatorname{Hom}(A, B)$.

For each object $A$ there is a morphism $1_A: A \rightarrow A$ called the identity morphism on $A$. (When $A$ can be determined by context, its denoted simply 1.)

Given two morphisms $A \stackrel{f}{\rightarrow} B$ and $B \stackrel{g}{\rightarrow} C$ where the codomain of one is the same as the domain of the other there is another morphism $A \stackrel{g \circ f}{\longrightarrow} C$ called the composition of the two morphisms. This composition is illustrated by the commutative diagram For all $A \stackrel{f}{\rightarrow} B, B \stackrel{g}{\rightarrow} C$, and $C \stackrel{h}{\rightarrow} D,(h \circ g) \circ f=h \circ(g \circ f)$. In the diagram below, if the two triangles in the diagram each commute, then the parallelogram commutes.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|The field of rational numbers, fields of fractions

$$(m, n) \equiv(k, l) \quad \text { iff } \quad m l=n k$$

## 数学代写|现代代数代考Modern Algebra代写|The formal definition of categories

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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。