### 数学代写|代数数论代写Algebraic number theory代考|Minkowski’s Lemma on Convex Bodies

statistics-lab™ 为您的留学生涯保驾护航 在代写代数数论Algebraic number theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写代数数论Algebraic number theory代写方面经验极为丰富，各种代写代数数论Algebraic number theory相关的作业也就用不着说。

• 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|Minkowski’s Lemma on Convex Bodies

The Dirichlet’s unit theorem asserts that, up to the roots of unity in $K$, the group $\mathcal{O}{K}^{\times}$of units of $K$ is a free Abelian group of rank $r=r{1}+r_{2}-1$. [We shall define the non-negative integers $r_{1}$ and $r_{2}$ in the next section.] It is not very difficult to show that $r \leq r_{1}+r_{2}-1$. The harder part that $r=r_{1}+r_{2}-1$ follows from the famous lemma of Minkowski on convex bodies.

A subset $X \subseteq \mathbb{R}^{n}$ is convex if for all $\boldsymbol{u}, \boldsymbol{v}$ in $X$ and all real $t$ in the interval $[0,1]$, the vector $t \boldsymbol{u}+(1-t) \boldsymbol{v}$ is in $X$. That is, the line segment joining $\boldsymbol{u}$ to $\boldsymbol{v}$ is entirely in $X$. It is easy to see that if $X$ is convex in $\mathbb{R}^{m}$ and $Y$ is convex in $\mathbb{R}^{n}$, then $X \times Y$ is convex in $\mathbb{R}^{m+n}$. We call $X \subseteq \mathbb{R}^{n}$ centrally symmetric if $\boldsymbol{v} \in X$ implies $-\boldsymbol{v} \in X$.

Let $\mu$ be the Lebesgue measure on $\mathbb{R}^{n}$, that is, the measure on $\mathbb{R}^{n}$, such that for a cube $X \subseteq \mathbb{R}^{n}$ given by
$$\begin{gathered} X=\left{\boldsymbol{x}=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n} \mid a_{j} \leq x_{j} \leq b_{j}\right} \ \mu(X)=\operatorname{vol}(X)=\prod_{j=1}^{n}\left(b_{j}-a_{j}\right) \end{gathered}$$
Let $L$ be a full lattice with a fundamental parallelepiped $P$, as in (5.4) and (5.5). Of course, $P$ depends on the choice of the $\mathbb{Z}$-basis $\left{\boldsymbol{v}{1}, \ldots, \boldsymbol{v}{n}\right}$ of $L$. However, any two $\mathbb{Z}$-bases of $L$ are related by a unimodular matrix, that is a matrix of determinant $\pm 1$ with entries in $\mathbb{Z}$. Since $\mu(P)$ is the absolute value of the determinant, whose rows are $\boldsymbol{v}{1}, \ldots, \boldsymbol{v}{n}$, it follows that the volume $\mu(P)$ of $P$ is independent of the choice of the basis. Thus, we may denote $\mu(P)$ also by $\mu(L)$.

Theorem $5.4$ (Minkowski’s Lemma). Suppose $X \subseteq \mathbb{R}^{n}$ is a bounded, centrally symmetric convex set and $L \subseteq \mathbb{R}^{n}$ is a full lattice. If $\mu(X)>2^{n} \mu(L)$, then $X$ contains a nonzero vector of $L$.

Proof. First we show that if $Y \subseteq \mathbb{R}^{n}$ is a bounded set, such that ${\boldsymbol{v}+Y \mid \boldsymbol{v} \in$ $L}$ is a family of disjoint subsets of $\mathbb{R}^{n}$, then $\mu(Y) \leq \mu(P)$, where $P$ is a fundamental parallelepiped of $L$. This is almost immediate, because writing $Y$ as the disjoint union
$$Y=\cup_{v \in L} Y \cap(v+P),$$
we have by $(5.6), \mu(Y)=\sum_{v \in L} \mu(Y \cap(v+P))$.
Since $\mu$ is translation invariant, $\mu(Y \cap(v+P))=\mu((-v+Y) \cap P)$. Hence $\mu(Y)=\sum_{v \in L} \mu((-v+Y) \cap P) \leq \mu(P)$, because the sets $-v+Y$ are also pairwise disjoint.

## 数学代写|代数数论代写Algebraic number theory代考|Logarithmic Embedding

Suppose $K \subseteq \mathbb{C}$ is a number field of degree $n$ over $\mathbb{Q}$. Consider a ring homomorphism $\sigma: K \rightarrow \mathbb{C}$. We require that $\sigma(1)=1$. Hence $\sigma_{\mid \mathbb{Q}}=1_{\mathrm{Q}}$, the identity map on $\mathbb{Q}$. Such a $\sigma$ is clearly injective. [Its kernel Ker $(\sigma)$ is an ideal of the field $K$, which can only be ${0}$ or $K$.] Hence, we call $\sigma$ a Q-isomorphism of $K$ into $\mathbb{C}$. There are exactly $n \mathbb{Q}$-isomorphisms of $K$ into $\mathbb{C}$. To see this, write $K=\mathbb{Q}(\alpha)$. If $\sigma$ is a $\mathbb{Q}$-isomorphism of $K$ into $\mathbb{C}$, it is determined by $\sigma(\alpha)$, which is a conjugate of $\alpha$. But there are exactly $n$ conjugates of $\alpha$ over $\mathbb{Q}$.
One may regard such a $\sigma: K \rightarrow \mathbb{C}$ also an injective linear transformation of vector spaces, when $K$ and $\mathbb{C}$ are viewed as vector spaces over $\mathbb{Q}$. Unless stated to the contrary $\sigma: K \rightarrow \mathbb{C}$ will be a $\mathbb{Q}$-isomorphism.

If $\sigma(K) \subseteq \mathbb{R}$, we call $\sigma$ a real imbedding, otherwise it is a complex imbedding. If $\sigma$ is complex, the map $\bar{\sigma}: K \rightarrow \mathbb{C}$, given by $\bar{\sigma}(x)=\overline{\sigma(x)}$ is also a $\mathbb{Q}$ isomorphism. Thus, the complex $\mathbb{Q}$-isomorphisms occur in pairs. We shall denote the real $\mathbb{}$ complex ones by $\sigma_{r_{1}+1}, \overline{\sigma_{r_{1}+1}}, \ldots ; \sigma_{r_{1}+r_{2}}, \overline{\sigma_{r_{1}+r_{2}}}$. In particular, $n=r_{1}+2 r_{2}$.
Consider $\mathbb{C}$ as a vector space of dimension two over $\mathbb{R}$ with ${1, i}$ as the standard basis. If $z=x+i y \in \mathbb{C}$, the multiplication by $z$ is a linear transformation of $\mathbb{C}$ into itself over $\mathbb{R}$. Its matrix relative to the basis ${1, i}$ is easily seen to be $T=\left(\begin{array}{cc}x & y \ -y & x\end{array}\right)$ with determinant
$$\operatorname{det}(T)=x^{2}+y^{2}=|z|^{2}$$
If we identify $\mathbb{C}$, as a vector space over $\mathbb{R}$ with $\mathbb{R}^{2}$, via the map $x+i y \rightarrow\left(\begin{array}{l}x \ y\end{array}\right)$, then $\mathbb{R}^{r_{1}} \times \mathbb{C}^{r_{2}} \cong \mathbb{R}^{n}$.

## 数学代写|代数数论代写Algebraic number theory代考|Units of a Quadratic Field

Let $K=\mathbb{Q}(\sqrt{d})(d \neq 0,1$, a square-free integer $)$ be a quadratic field. We call $K$ a real quadratic field or an imaginary quadratic field according as $d>0$ or $d<0$. If $K$ is an imaginary quadratic field, then $r_{1}=0, r_{2}=1$, so $r=r_{1}+r_{2}-1=0$. In this case, $\mathcal{O}{K}^{\times}=W{K}$, the roots of unity in $K$. We leave it as an exercise to determine this finite group $W_{K}$.

For the real quadratic field, $r=1$ and the group of units is given by the following corollary.

Corollary 5.15. If $d>1$ is a square-free integer and $K=\mathbb{Q}(\sqrt{d})$, then the group
$$\mathcal{O}_{K}^{\times} \cong{\pm 1} \times \mathbb{Z} .$$
In particular, the Pell equation $x^{2}-d y^{2}=1$ has infinitely many solutions in integers.
EXERCISES

1. Determine the structure of $\mathcal{O}_{K}^{\times}$when $[K: \mathbb{Q}]=3$ and 4 .
2. Use Dirichlet’s unit theorem to find all integer solutions of $5 x^{2}-$ $5 y^{2}=y^{4}$.

## 数学代写|代数数论代写Algebraic number theory代考|Minkowski’s Lemma on Convex Bodies

\begin{聚集} X=\left{\boldsymbol{x}=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n} \mid a_{j } \leq x_{j} \leq b_{j}\right} \ \mu(X)=\operatorname{vol}(X)=\prod_{j=1}^{n}\left(b_{j} -a_{j}\right) \end{聚集}\begin{聚集} X=\left{\boldsymbol{x}=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n} \mid a_{j } \leq x_{j} \leq b_{j}\right} \ \mu(X)=\operatorname{vol}(X)=\prod_{j=1}^{n}\left(b_{j} -a_{j}\right) \end{聚集}

## 数学代写|代数数论代写Algebraic number theory代考|Units of a Quadratic Field

○ķ×≅±1×从.

1. 确定结构○ķ×什么时候[ķ:问]=3和 4 。
2. 使用狄利克雷单位定理求所有整数解5X2− 5是2=是4.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。