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# 数学代写|代数数论代写Algebraic Number Theory代考|MATH661 Integral Domains

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## 数学代写|代数数论代写Algebraic Number Theory代考|Integral Domains

A nonzero element $a$ of a ring $A$ (always commutative) is called a zero divisor if $a b=0$ for a nonzero $b$ in $A$. In the ring $\mathbb{Z} / 6 \mathbb{Z}, 2,3$, and 4 are the only divisors of zero. A field has no divisor of zero. A ring without zero divisors is called an integral domain or simply a domain. We have already discussed many integral domains which are not fields, e.g. $\mathbb{Z}, \mathbb{Z}[i], \mathbb{Z}[\omega]$ and $\mathbb{Z}[\sqrt{d}]$ for $d \neq 0$, a square-free integer, which are relevant to our subject.

An element $u$ in $A$ is a $u n i t$ if $u v=1$ for some $v$ in $B$. For example, the only units in the ring $\mathbb{Z}$ are $\pm 1$.

Definition 2.7. A domain $A$ is a Euclidean domain if there is a map which assigns to each nonzero element $\alpha$ of $A$ a non-negative integer $d(\alpha)$ such that for all nonzero $\alpha, \beta$ in $A$,
i) $d(\alpha) \leq d(\alpha \beta)$, and
ii) $A$ has elements $q$ (the quotient) and $\gamma$ (the remainder) so that $\alpha=q \beta+\gamma$ and either $\gamma=0$ or $d(\gamma)<d(\beta)$.

With the Euclidean algorithm, both $\mathbb{Z}$ and the ring $k[x]$ of polynomials over a field $k$ are Euclidean domains. For $\mathbb{Z}, d(\alpha)=|\alpha|$ and for $k[x], d(f(x))=$ $\operatorname{deg} f(x)$

## 数学代写|代数数论代写Algebraic Number Theory代考|Factoring Rational Primes in Zi

Let $A$ be the ring $\mathbb{Z}[i]$ of Gaussian integers and $p=2,3,4, \ldots$ a rational prime. This $p$ may or may not be a prime element of $A$. To find exactly when it is, recall the famous theorem of Fermat on the sum of two squares, which was proved by Euler (cf. [8, p. 48]).

Theorem $2.14$ (Fermat). An odd prime $p$ in $\mathbb{Z}$ is a sum of two squares $\left(p=a^2+b^2\right)$ if and only if $p=4 k+1$ for $k$ in $\mathbb{N}$.

The norm of any divisor of $\alpha=a+i b$ must be a divisor of $N(\alpha)=a^2+b^2$, and for $\alpha=\beta \gamma$ with $\beta$, $\gamma$ both non-units, $1<N(\beta)<N(\alpha)$ (only the units have norm 1). Therefore, if $a^2+b^2$ is a prime, then $\alpha$ has to be a prime in $\mathbb{Z}[i]$. We have thus proved the following fact:

Theorem 2.15. A prime $p$ is a sum of two squares, $p=a^2+b^2 \Leftrightarrow p$ is $a$ product $(a+i b)(a-i b)$ of two primes $a \pm i b$ in $\mathbb{Z}[i]$.

For $p=2$, its two prime factors $1+i, 1-i$ in $\mathbb{Z}[i]$ are associates: $1+i=i(1-i)$. Therefore,
$$2=i(1-i)^2 .$$
We say that 2 ramifies in $\mathbb{Z}[i]$. By Fermat’s Theorem (Theorem 2.15), $p \equiv 1$ $(\bmod 4) \Leftrightarrow p$ is a product
$$p=\pi_1 \pi_2$$
of two primes $\pi_1, \pi_2$ in $\mathbb{Z}[i]$. Moreover, $\pi_1$ and $\pi_2$ are complex conjugates of each other and hence they are distinct. This discussion can be wrapped up as follows: In order to do that, observe that ${1, i}$ is a $\mathbb{Z}$-bases of $\mathbb{Z}[i]$ and so is its conjugate ${1,-i}$. These two bases make a $2 \times 2$ matrix
$$A=\left(\begin{array}{cc} 1 & i \ 1 & -i \end{array}\right)$$
with $|\operatorname{det}(A)|=2$, called the discriminant of $\mathbb{Q}(i)$.

## 数学代写|代数数论代写代数数论代考|积分域

i) $d(\alpha) \leq d(\alpha \beta)$和
ii) $A$中有元素$q$(商)和$\gamma$(余数)，那么$\alpha=q \beta+\gamma$和$\gamma=0$或$d(\gamma)<d(\beta)$

$$p=\pi_1 \pi_2$$
。此外，$\pi_1$和$\pi_2$是彼此的复共轭，因此它们是不同的。这个讨论可以总结如下:为了做到这一点，观察${1, i}$是$\mathbb{Z}[i]$的$\mathbb{Z}$ -bases，它的共轭${1,-i}$也是。这两个基底构成一个$2 \times 2$矩阵
$$A=\left(\begin{array}{cc} 1 & i \ 1 & -i \end{array}\right)$$

## 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。