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# 数学代写|数论代写Number Theory代考|Math204A Historical Background

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## 数学代写|数论代写Number Theory代考|Historical Background

The origin of Algebraic Number theory is attributed to Fermat’s Last Theorem which was conjectured by a French mathematician Pierre de Fermat in 1637. It states that the equation $X^n+Y^n=Z^n$ has no solution in non-zero integers $x, y, z$, when $n$ is an integer greater than 2 . Fermat himself proved the case $n=4$ of the theorem (see [Ded2, 0.3.1]). If $n=p m$, then the relation $x^n+y^n=z^n$ implies that $\left(x^m\right)^p+$ $\left(y^m\right)^p=\left(z^m\right)^p$ which gives a solution of the equation $X^p+Y^p=Z^p$. Since any integer greater than 2 is either a multiple of 4 or has an odd prime factor, for proving Fermat’s Last Theorem it is enough to show that $X^p+Y^p=Z^p$ has no solution in non-zero integers for all odd prime exponents $p$. This celebrated theorem motivated a general study of the theory of algebraic numbers. History reveals that in 1770 , Leonhard Euler used the field $\mathbb{Q}(\omega)$ with $\omega$ a complex cube root of unity to prove Fermat’s Last Theorem for the case $n=3$ (cf. [Ded2, 0.5.1]). The first major step towards a general proof of Fermat’s Last Theorem was by a French woman ${ }^1$ Sophie Germain. In a letter dated May 12,1819 to the greatest number theorist of that time Carl Friedrich Gauss, she explained her idea of the proof. She had proved that if $p$ is an odd prime such that $q=2 k p+1$ is also a prime for some number $k$ satisfying the following conditions: (i) $x^p \equiv p(\bmod q)$ has no solution (ii) the set of $p$ th powers modulo $q$ contains no consecutive non-zero integers, then the first case of Fermat’s Last Theorem holds for the exponent $p$, i.e., the equation $X^p+Y^p=Z^p$ has no solution in integers $x, y, z$ with $p$ not dividing $x y z$. In particular, for an odd prime $p$ if $2 p+1$ is also a prime, then the first case of Fermat’s Last Theorem holds for the exponent $p$. In this way she was able to show that the same holds for all odd primes $p \leq 197$. In 1825 , her method claimed its first complete success when the famous mathematicians Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre (one German and the other French) working independently were able to prove the case $n=5$ of Fermat’s Last Theorem. In fact, they acknowledged that their proofs were based on the method of Sophie Germain. Fourteen years later, the French mathematician Gabriel Lamé proved the case $n=7$ of the theorem using Germain’s results. Her results related to Fermat’s Last Theorem remained most important until the contribution of Eduard Kummer in $1847 .$

## 数学代写|数论代写Number Theory代考|Algebraic Numbers and Algebraic Integers

We begin by introducing some basic notions of algebraic number theory.
Definition A complex number $\alpha$ is said to be an algebraic number if $\alpha$ is a root of a non-zero polynomial with coefficients from the field $\mathbb{Q}$ of rational numbers. A complex number which is not an algebraic number is called a transcendental number.
Note that if $\alpha$ is an algebraic number, then the degree of the extension $\mathbb{Q}(\alpha)$ over $\mathbb{Q}$ is finite and vice versa.

Theorem 1.1 The set of all algebraic numbers is a subfield of $\mathbb{C}$, the field of complex numbers.

Proof Suppose that $\alpha, \beta$ are algebraic numbers with $\beta \neq 0$. We have to show that $\alpha \pm \beta, \alpha \beta$ and $\frac{\alpha}{\beta}$ are algebraic numbers. The extensions $\mathbb{Q}(\alpha) / \mathbb{Q}$ and $\mathbb{Q}(\beta) / \mathbb{Q}$ are finite, say of degree $m$ and $n$ respectively. Since
$$[\mathbb{Q}(\alpha, \beta): \mathbb{Q}(\alpha)] \leq[\mathbb{Q}(\beta): \mathbb{Q}]=n,$$
it follows from Tower theorem (cf. Theorem A.1) that
$$[\mathbb{Q}(\alpha, \beta): \mathbb{Q}]=[\mathbb{Q}(\alpha, \beta): \mathbb{Q}(\alpha)][\mathbb{Q}(\alpha): \mathbb{Q}] \leq m n .$$
As the elements $\alpha \pm \beta, \alpha \beta$ and $\frac{\alpha}{\beta}$ belong to $\mathbb{Q}(\alpha, \beta)$, therefore the degree of the extension obtained by adjoining any of these elements to $\mathbb{Q}$ is finite and hence the theorem is proved.

## 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。