### 数学代写|代数数论代写Algebraic number theory代考|Review of the Prerequisite Material

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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|代数数论代写Algebraic number theory代考|Basic Concepts

A group is a pair $(G, )$ of a nonempty set $G$ and a binary operation $$on G, i.e. a map G \times G \ni(x, y) \rightarrow x * y \in G, called the group law on G with the following properties: i) The group law is associative: for all x y, z in G,(x * y) * z=x *(y * z), ii) there is an element e in G, called the identity, such that e * x=x * e=x for all x in G and iii) for each x in G there is a y in G, such that x * y=y * x=e. We denote y by x^{-1}, the inverse of x. We call the group (G, *) Abelian if for all x, y in G, x * y=y * x. In this case * is usually denoted by +x^{-1} by -x, and e by 0 . We call -x the additive inverse of x. Often the product x * y is written simply as x y and x^{-1} is called the multiplicative inverse of x. It turns out that e and x^{-1} are unique. The most familiar examples of Abelian groups are (G,+) with G=\mathbb{Z}, \mathbb{Q}, \mathbb{R} and \mathbb{C}. An example of a nonAbelian group is the general linear group G L(n, \mathbb{Z}) of n \times n matrices with integer entries and determinant \pm 1 under matrix multiplication. A ring is set A with at least two distinct elements, denoted by 0 and 1 having two binary operations (addition and multiplication) such that i) (A,+) is an Abelian group with 0 as its identity, ii) 1 x=x 1=x for all x in A and iii) the multiplication is associative and distributive over the addition:$$
x(y+z)=x y+x z \text { and }(x+y) z=x z+y z .
$$## 数学代写|代数数论代写Algebraic number theory代考|Galois Extensions We assume all our fields to be subfields of \mathbb{C}. Let K / k be a field extension. The set \mathrm{Gal}(K / k) of the field automorphisms \sigma of K such that \sigma(a)=a for all a in k is a (usually non-Abelian) group under the composition of maps. It is called the Galois group of K over k. In general, for a finite extension K / k, |\operatorname{Gal}(K / k)| \leq[K: k]. We call K / k a Galois extension if the equality holds. Examples of Galois Groups First, let K / k be any field extension, not necessarily finite. Let \alpha in K be a root of a polynomial$$
f(x)=c_{0}+c_{1} x+\cdots+c_{n} x^{n}
$$over k. If \sigma \in \operatorname{Gal}(K / k), then$$
\begin{aligned}
f(\sigma(\alpha)) &=c_{0}+c_{1} \sigma(\alpha)+\cdots+c_{n}(\sigma(\alpha))^{n} \
&=\sigma(f(\alpha))=\sigma(0)=0
\end{aligned}
$$Thus \sigma(\alpha) is also a root of f(x). This simple observation will be crucial to what follows. Let K be a quadratic field, a field extension of \mathbb{Q} of degree 2 . Then one checks that (Exercise 16 ) K=\mathbb{Q}(\sqrt{d})={r+s \sqrt{d} \mid r, s \in \mathbb{Q}} for a square-free integer d \neq 0,1. Example 2.1. Let us take d=-1. There are exactly two automorphisms of K whose restrictions to \mathbb{Q} is the identity map on Q. The identity map 1 on K itself and \sigma which takes i to its conjugate, the other root -i of x^{2}+1. Thus \operatorname{Gal}(K / k) \cong{\pm 1} and Q(i) is a Galois extension of \mathbb{Q}. Example 2.2. Now take d=-3. Then \mathbb{Q}(\omega)={r+s \omega \mid r, s \in \mathbb{Q}}. The Galois group \operatorname{Gal}(K / k) consists of two elements, the identity automorphism 1 of K and the automorphism \sigma of K such that \sigma(\omega)=\bar{\omega}. [Note that \bar{\omega}=\omega^{2}=\frac{1}{\omega}.] Hence \mathbb{Q}(\omega) / \mathbb{Q} is also an Abelian extension. Example 2.3. Let \alpha be the real cube root of 2, \alpha=\sqrt[3]{2}, K=\mathbb{Q}(\alpha) the smallest subfield of \mathbb{C} containing \alpha. The other cube roots of 2 which are \omega \alpha and \omega^{2} \alpha are not in K. Thus there is only one element in the Galois group \operatorname{Gal}(K / \mathbb{Q}), namely the identity element of the group \operatorname{Gal}(K / \mathbb{Q}). Since [K: \mathbb{Q}]=3 but |\operatorname{Gal}(K / \mathbb{Q})|=1, the extension K / \mathbb{Q} is not Galois. The following is a standard result in field theory: Theorem 2.4. If K / k is a field extension of degree d, then there is an \alpha in K such that 1, \alpha, \alpha^{2}, \ldots, \alpha^{d-1} is a basis of K as a vector space over k. In fact, \alpha is a root of an irreducible polynomial f(x) over k of degree d. Definition 2.5. If all the d roots of this f(x) are in K, we call the extension K / k normal. Remark 2.6. i) According to our definition of the Galois extension, an extension is normal if and only if it is Galois. ii) The Galois group Gal (K / k) is often defined only for normal extensions, in which case \operatorname{Gal}(K / k) is always equal to the degree [K: k] of the field extension K / k. ## 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 unit 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). ## 代数数论代考 ## 数学代写|代数数论代写Algebraic number theory代考|Basic Concepts 一组是一对(G,)非空集G和一个二元运算$$ onG，即地图G×G∋(X,是)→X∗是∈G，称为群定律G具有以下性质：
i) 群律是结合的：对于所有X是,和在G,(X∗是)∗和=X∗(是∗和),
ii) 有一个元素和在G，称为恒等式，这样和∗X=X∗和=X对所有人X在Giii
) 对于每个X在G有一个是在G, 这样X∗是=是∗X=和.

i)(一个,+)是一个以 0 为恒等式的阿贝尔群，

ii)1X=X1=X对所有人X在一个
iii) 乘法对加法是关联和分布的：

X(是+和)=X是+X和 和 (X+是)和=X和+是和.

## 数学代写|代数数论代写Algebraic number theory代考|Galois Extensions

F(X)=C0+C1X+⋯+CnXn

F(σ(一个))=C0+C1σ(一个)+⋯+Cn(σ(一个))n =σ(F(一个))=σ(0)=0

ii) 伽罗瓦群 Gal(ķ/ķ)通常只为正常扩展定义，在这种情况下加尔⁡(ķ/ķ)总是等于度数[ķ:ķ]字段扩展ķ/ķ.

## 数学代写|代数数论代写Algebraic number theory代考|Integral Domains

ii)一个有元素q（商）和C（余数）这样一个=qb+C并且要么C=0或者d(C)<d(b).

## 有限元方法代写

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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。