Posted on Categories:Modern Algebra, 数学代写, 现代代数

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## 数学代写|现代代数代考Modern Algebra代写|The Gaussian integers, Z[i]

One important example of an integral domain is that of the Gaussian integers $\mathbf{Z}[i]$. Its elements are of the form $x+y i$ where $x, y \in \mathbf{Z}$, so they can be viewed as a lattice of points in the complex plane as in figure 3.1. You can check that $\mathbf{Z}[i]$ is closed under addition, subtraction, multiplication, and includes 1 , so it is a subring of the field C. Therefore, it’s an integral domain. We’ll see later that $\mathbf{Z}[i]$ is a particularly nice integral domain called a Euclidean domain.

There are four units (elements having reciprocals) in the Gaussian integers. Besides 1 and $-1, i$ and $-i$ are also units. Note that $(1+i)(1-i)=2$, so 2 is not prime in $\mathbf{Z}[i]$ even though it is prime in $\mathbf{Z}$.

We’ll come back to $\mathbf{Z}[i]$ when we study Euclidean domains in section 3.8.4. Also $\mathbf{Z}[i]$ is an example of a “ring of integers” to be defined in section $3.11$.

Eisenstein integers. The Eisenstein integers are similar to the Gaussian integers, but instead of consisting of a square lattices of complex numbers, they consist of a triangular lattice of complex numbers. They include complex numbers of the form $z=x+y \omega$ where $\omega$ is the cube root of $1, \omega=\frac{1}{2}(-1+i \sqrt{3})=e^{2 \pi i / 3}$. See figure $3.3$ for the lattice of Eisenstein integers.

## 数学代写|现代代数代考Modern Algebra代写|Finite fields again

We won’t find any examples of finite integral domains that aren’t fields because there aren’t any.
Theorem 3.7 (Wedderburn). If $R$ is a finite integral domain, then $R$ is a field.

Proof. Let $x$ be a nonzero element of $R$. Consider the positive powers of $x$ :
$$x, x^2, x^3, \ldots, x^n \ldots$$
Since there are infinitely many powers, but only finitely many elements in $R$, therefore at least two distinct powers are equal. Let, then, $x^m=x^n$ with $m<n$. Cancel $x^m$ from each side of the equation (which is possible because $R$ is an integral domain) to conclude $x^{n-m}=1$. Therefore, the reciprocal of $x$ is $x^{n-m-1}$. Therefore, every nonzero element has an inverse.

Q.E.D.
This theorem can be used to give a short proof that $\mathbf{Z}_p$ is a field when $p$ is a prime, since it’s easy to show that $\mathbf{Z}_p$ is an integral domain. We’ll show it has no zero-divisors. Suppose that $x y \equiv 0(\bmod p)$. Then $p \mid x y$. But if a prime divides a product, it divides one of the factors, so either $p \mid x$ or $p \mid y$, in other words, either $x \equiv 0(\bmod p)$ or $y \equiv 0(\bmod p)$. Thus, $\mathbf{Z}_p$ is an integral domain, and hence, by the above theorem, it’s a field.

Our earlier, more complicated proof used the extended Euclidean algorithm to find an inverse for $x$. That’s actually a much more efficient way to find the inverse than to look through the powers of $x$.

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|Finite fields again

$$x, x^2, x^3, \ldots, x^n \ldots$$

QED

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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。