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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|Closure properties of algebraic numbers

Next we shall sketch proofs that the set of (complex) algebraic numbers forms a subfield of $\mathbb{C}$, and that the algebraic integers form an integral domain. These proofs require a certain acquaintance with basic properties of vector spaces and abelian groups; however, the level required is probably too much to summarise in an appendix. Therefore, on this occasion only, we invite the interested reader to refer to other sources for background material. Two of many possibilities are Axler [8] for linear algebra, Stewart and Tall [62] for groups. The reader who prefers to continue with the main topics of this book can safely proceed to section $3.2$ after noting carefully the results of Theorem 3.10, Corollary $3.11$ and Theorem 3.12.
Lemma 3.8. Let $S=\left{\alpha_{k} \mid k \in K\right}$ be a set of complex numbers. Then

• the set of linear combinations
$$\sum r_{k} \alpha_{k}$$
with finitely many terms and rational coefficients $r_{k}$ is a vector space over the field $\mathbb{Q}$;
• the set of linear combinations
$$\sum m_{k} \alpha_{k}$$
with finitely many terms and integer coefficients $m_{k}$ is an abelian group under addition.

Lemma 3.9. Finiteness criteria for algebraic numbers. Let $\alpha \in \mathbb{C}$; in the previous lemma take $S=\left{1, \alpha, \alpha^{2}, \ldots\right}$. Then

• $\alpha$ is algebraic if and only if the vector space of rational linear combinations of $S$ is finite-dimensional;
• $\alpha$ is an algebraic integer if and only if the group of integer linear combinations of $S$ is finitely generated.

## 数学代写|数论作业代写number theory代考|EXISTENCE OF TRANSCENDENTAL NUMBERS

The first question we need to address about transcendental numbers is whether or not there are any! It is clear that algebraic numbers exist: for a start, all rational numbers are algebraic, and we have also given a few examples of irrational algebraic numbers. However, it is conceivable that every complex number could be a root of a rational polynomial, in which case transcendental numbers would not exist.

Notice, by the way, that we have so far only seen algebraic numbers of degree up to 4 . It is not at all clear that algebraic numbers of arbitrarily high degree exist. If, for example, we were to consider polynomials with real (rather than rational) coefficients, then there would be no irreducible polynomials of degree greater than 2. The situation in this case would therefore be very simple: all real numbers would be algebraic (over $\mathbb{R}$ ) of degree 1 , and all nonreal complex numbers would be algebraic (over $\mathbb{R}$ ) of degree 2. Among the complex numbers there would be no algebraic numbers of higher degree, and no transcendental numbers.

The existence of transcendental numbers was first proved by Joseph Liouville, who attempted to show that $e$ is not an algebraic number. He failed in this aim but achieved enough to allow him in 1844 (and again, using different techniques, in 1851) to give specific examples of transcendental numbers. A completely different proof was given three decades later by Georg Cantor: a proof which is perhaps simpler, though, as it does not provide any specific examples of transcendentals, possibly somehow beside the point as far as number theory is concerned. We shall begin with Cantor’s proof.

Cantor proved the existence of transcendental numbers simply by showing that there are, in a sense, more complex numbers than algebraic numbers. Specifically, the set of complex numbers is uncountable – this follows immediately from the uncountability of the reals, proved by Cantor in 1874 – while, as we shall now show, the set of (complex) algebraic numbers is countable.
First, a slightly informal proof. Recall that an algebraic number is, (almost) by definition, a root of a non-zero polynomial with integral coefficients. Define the height of any such polynomial to be the maximum of the absolute values of its coefficients: that is, if $f(z)=a_{n} z^{n}+a_{n-1} z^{n-1}+\cdots+a_{1} z+a_{0}$ with all $a_{k}$ integers and $a_{n} \neq 0$, then
$$H(f)=\max \left(\left|a_{n}\right|,\left|a_{n-1}\right|, \ldots,\left|a_{1}\right|,\left|a_{0}\right|\right)$$

## 数学代写|数论作业代写number theory代考|Closure properties of algebraic numbers

• 线性组合的集合
$$\sum r_{k} \alpha_{k}$$
具有有限多项和有理系数 $r_{k}$ 是场上的向量空间 $\mathbb{Q}$;
• 线性组合的集合
$$\sum m_{k} \alpha_{k}$$
具有有限多项和整数系数 $m_{k}$ 是加法下的阿贝尔群。
引理 3.9。代数数的有限性准则。让 $\alpha \in \mathbb{C}$; 在前面的引理中 $\backslash 1$ ef $\mathrm{t}$ 的分隔符缺失或无法识别 .然后
• $\alpha$ 是代数的当且仅当 $S$ 是有限维的；
• $\alpha$ 是一个代数整数当且仅当整数线性组合的群 $S$ 是有限生成的。

## 数学代写|数论作业代写number theory代考|EXISTENCE OF TRANSCENDENTAL NUMBERS

$$H(f)=\max \left(\left|a_{n}\right|,\left|a_{n-1}\right|, \ldots,\left|a_{1}\right|,\left|a_{0}\right|\right)$$

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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。

assignmentutor™您的专属作业导师
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