Posted on Categories:Algebraic Number Theory, 代数数论, 数学代写

# 数学代写|代数数论代写Algebraic Number Theory代考|MA58400 Primes in Special Progressions

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|代数数论代写Algebraic Number Theory代考|Primes in Special Progressions

Another interesting application of quadratic reciprocity is that it can be used to show there exist infinitely many primes in certain arithmetic progressions. In the next two exercises, we imitate Euclid’s proof for the existence of an infinite number of primes to show that there are infinitely many primes in the following two arithmetic progressions, $4 k+1$ and $8 k+7$.
Exercise 7.5.1 Show that there are infinitely many primes of the form $4 k+1$.
Exercise 7.5.2 Show that there are infinitely many primes of the form $8 k+7$.
The results we have just derived are just a special case of a theorem proved by Dirichlet. Dirichlet proved that if $l$ and $k$ are coprime integers, then there must exist an infinite number of primes $p$ such that $p \equiv l$ $(\bmod k)$. What is interesting about these two exercises, however, is the fact that we used a proof similar to Euclid’s proof for the existence of an infinite number of primes. An obvious question to ask is whether questions about all arithmetic progressions can be solved in a similar fashion.

The answer, sadly, is no. However, not all is lost. It can be shown that if $l^2 \equiv 1(\bmod k)$, then we can apply a Euclid-type proof to show there exist an infinite number of primes $p$ such that $p \equiv l(\bmod k)$. (See Schur [S]. For instance, Exercises 1.2.6 and 5.6.10 give Euclid-type proofs for $p \equiv 1$ $(\bmod k)$ using cyclotomic polynomials.) Surprisingly, the converse of this statement is true as well. The proof is not difficult, but involves some Galois Theory. It is due to Murty $[\mathrm{Mu}]$.
We can restate our two previous exercises as follows:
(1) Are there infinitely many primes $p$ such that $p \equiv 1(\bmod 4)$ ?
(2) $p \equiv 7(\bmod 8)$ ?
From what we have just discussed, we observe that we can indeed apply a Euclid-type proof since $1^2 \equiv 1(\bmod 4)$ and $7^2 \equiv 1(\bmod 8)$.

## 数学代写|代数数论代写Algebraic Number Theory代考|Dirichlet’s Unit Theorem

Let $K$ be a number field and $\mathcal{O}_K$ its ring of integers. An element $\alpha \in \mathcal{O}_K$ is called a unit if $\exists \beta \in \mathcal{O}_K$ such that $\alpha \beta=1$. Evidently, the set of all units in $\mathcal{O}_K$ forms a multiplicative subgroup of $K^*$, which we will call the unit group of $K$.

In this chapter, we will prove the following fundamental theorem, which gives an almost complete description of the structure of the unit group of $K$, for any number field $K$.

Theorem (Dirichlet’s Unit Theorem) Let $U_K$ be the unit group of $K$. Let $n=[K: \mathbb{Q}]$ and write $n=r_1+2 r_2$, where, as usual, $r_1$ and $2 r_2$ are, respectively, the number of real and nonreal embeddings of $K$ in $\mathbb{C}$. Then there exist fundamental units $\varepsilon_1, \ldots, \varepsilon_r$, where $r=r_1+r_2-1$, such that every unit $\varepsilon \in U_K$ can be written uniquely in the form
$$\varepsilon=\zeta \varepsilon_1^{n_1} \cdots \varepsilon_r^{n_r},$$
where $n_1, \ldots, n_r \in \mathbb{Z}$ and $\zeta$ is a root of unity in $\mathcal{O}_K$. More precisely, if $W_K$ is the subgroup of $U_K$ consisting of roots of unity, then $W_K$ is finite and cyclic and $U_K \simeq W_K \times \mathbb{Z}^r$.

## 数学代写|代数数论代写Algebraic Number Theory代考|Primes in Special Progressions

(1) 是否存在无穷多个溸数 $p$ 这样 $p \equiv 1(\bmod 4)$ ?
(2) $p \equiv 7(\bmod 8)$ ?

## 数学代写|代数数论代写Algebraic Number Theory代考|Dirichlet’s Unit Theorem

$$\varepsilon=\zeta \varepsilon_1^{n_1} \cdots \varepsilon_r^{n_r},$$

## 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。