19th Ave New York, NY 95822, USA

# 数学代写|数论代写Number Theory代考|MATH453

my-assignmentexpert™提供最专业的一站式服务：Essay代写，Dissertation代写，Assignment代写，Paper代写，Proposal代写，Proposal代写，Literature Review代写，Online Course，Exam代考等等。my-assignmentexpert™专注为留学生提供Essay代写服务，拥有各个专业的博硕教师团队帮您代写，免费修改及辅导，保证成果完成的效率和质量。同时有多家检测平台帐号，包括Turnitin高级账户，检测论文不会留痕，写好后检测修改，放心可靠，经得起任何考验！

## 数学代写|数论代写Number Theory代考|Norm of a Prime Ideal

We begin by showing that each prime ideal in the ring of integers of an algebraic number field is associated with a unique rational prime.

Theorem 10.1.1 Let $K$ be an algebraic number field. Let $P$ be a prime ideal of $O_K$. Then there exists a unique rational prime $p$ such that
$$P \mid\langle p\rangle .$$
Proof: As $P$ is a prime ideal in $O_K, P \cap \mathbb{Z}$ is a prime ideal in $\mathbb{Z}$ (Theorem 1.6.2). Hence, by Theorems 1.4 .1 and 1.5.4, we have
$$P \cap \mathbb{Z}=\langle p\rangle$$
for some rational prime $p$. Thus
$$P \supseteq\langle p\rangle$$
and so by Theorem 8.4 .1 we have
$$P \mid\langle p\rangle .$$
Suppose $q$ is another rational prime such that
$$P \mid\langle q\rangle .$$
Then $P \supseteq\langle p\rangle$ and $P \supseteq\langle q\rangle$ so that
$$P \supseteq\langle p, q\rangle .$$
As $p$ and $q$ are distinct primes we have $\operatorname{gcd}(p, q)=1$ so that there are integers $a$ and $b$ such that $a p+b q=1$. Hence $1 \in\langle p, q\rangle \subseteq P$. Thus $O_K \subseteq P$, which is impossible.
Hence the prime $p$ is uniquely determined by $P \mid\langle p\rangle$.

## 数学代写|数论代写Number Theory代考|Factoring Primes in a Quadratic Field

Let $p$ be a rational prime and let $K$ be a quadratic field. By Theorem 10.1 .4 we have
$$g=g_K(p) \leq 2$$
so that
$$g=1 \text { or } 2 .$$
If $g=2$ by Theorem 10.1 .3 we have
$$e_1 f_1+e_2 f_2=2$$
so that
$$e_1=f_1=e_2=f_2=1 .$$
If $g=1$ we have
$$e_1 f_1=2$$
so that
$$\left(e_1, f_1\right)=(2,1) \text { or }(1,2) .$$
Thus in the case of a quadratic field there are just three possibilities:
(i) $g=2, e_1=f_1=e_2=f_2=1$,
(ii) $g=1, e_1=2, f_1=1$,
(iii) $g=1, e_1=1, f_1=2$.

In other words,
(i) $\langle p\rangle=P_1 P_2, N\left(P_1\right)=N\left(P_2\right)=p, P_1 \neq P_2$,
(ii) $\langle p\rangle=P^2, N(P)=p$,
(iii) $\langle p\rangle=P, N(P)=p^2$,
where $P_1, P_2, P$ denote prime ideals of $O_K$. In case (i) we say that $p$ splits in $K$, in case (ii) that $p$ ramifies in $K$, and in case (iii) that $p$ is inert (or remains prime) in $K$. In cases (i) and (iii) $p$ is unramified in $K$.

Our next theorem gives necessary and sufficient conditions for each of (i), (ii), (iii) to occur. As usual $\left(\frac{m}{p}\right)$ denotes the Legendre symbol of the integer $m$ modulo the odd prime $p$.

## 数学代写|数论代写Number Theory代考|Norm of a Prime Ideal

$$P \mid\langle p\rangle .$$

$$P \cap \mathbb{Z}=\langle p\rangle$$

$$P \supseteq\langle p\rangle$$

$$P \mid\langle p\rangle .$$

$$P \mid\langle q\rangle .$$

$$P \supseteq\langle p, q\rangle .$$

## 数学代写|数论代写Number Theory代考|Factoring Primes in a Quadratic Field

$$g=g_K(p) \leq 2$$

$$g=1 \text { or } 2 .$$

$$e_1 f_1+e_2 f_2=2$$

$$e_1=f_1=e_2=f_2=1 .$$

$$e_1 f_1=2$$

$$\left(e_1, f_1\right)=(2,1) \text { or }(1,2) .$$

(i) $g=2, e_1=f_1=e_2=f_2=1$;
(ii) $g=1, e_1=2, f_1=1$;
(iii) $g=1, e_1=1, f_1=2$。

(i) $\langle p\rangle=P_1 P_2, N\left(P_1\right)=N\left(P_2\right)=p, P_1 \neq P_2$;
(ii) $\langle p\rangle=P^2, N(P)=p$;
(iii) $\langle p\rangle=P, N(P)=p^2$;

## 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。