# 数学代写|数论作业代写number theory代考|Algebraic Integers in a Quadratic Field

#### Doug I. Jones

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## 数学代写|数论作业代写number theory代考|Algebraic Integers in a Quadratic Field

In this section we determine the algebraic integers in a field $\mathbb{Q}(\alpha)$ obtained by adjoining a root $\alpha(\in \mathbb{C})$ of an irreducible quadratic polynomial $x^2+a x+b \in \mathbb{Q}[x]$ to $\mathbb{Q}$; that is, $\mathbb{Q}(\alpha)$ is the smallest subfield of $\mathbb{C}$ containing both $\mathbb{Q}$ and $\alpha$. We note that $\alpha \notin \mathbb{Q}$ as $x^2+a x+b$ is irreducible in $\mathbb{Q}[x]$. Clearly
\begin{aligned} \mathbb{Q}(\alpha)= & \left{\frac{a_0+a_1 \alpha+\cdots+a_m \alpha^m}{b_0+b_1 \alpha+\cdots+b_n \alpha^n} \mid m, n\right. \text { (nonnegative integers), } \ & \left.a_0, \ldots, a_m, b_0, \ldots, b_n \in \mathbb{Q}, b_0+b_1 \alpha+\cdots+b_n \alpha^n \neq 0\right} . \end{aligned}
As $\alpha^2=-b-a \alpha$, we obtain recursively that $\alpha^k=c_k+d_k \alpha(k=2,3, \ldots)$, where $c_k, d_k \in \mathbb{Q}$. Thus
$$\mathbb{Q}(\alpha)=\left{\frac{e_0+e_1 \alpha}{f_0+f_1 \alpha} \mid e_0, e_1, f_0, f_1 \in \mathbb{Q},\left(f_0, f_1\right) \neq(0,0)\right} .$$
As $f_0^2-a f_0 f_1+b f_1^2 \neq 0$ for $\left(f_0, f_1\right) \neq(0,0)$ and
$$\frac{e_0+e_1 \alpha}{f_0+f_1 \alpha}=\left(\frac{e_0 f_0-a e_0 f_1+b e_1 f_1}{f_0^2-a f_0 f_1+b f_1^2}\right)+\left(\frac{e_1 f_0-e_0 f_1}{f_0^2-a f_0 f_1+b f_1^2}\right) \alpha,$$
we deduce that
$$\mathbb{Q}(\alpha)={x+y \alpha \mid x, y \in \mathbb{Q}},$$
where $\alpha^2+a \alpha+b=0$. The field $\mathbb{Q}(\alpha)$ is called a quadratic field or a quadratic extension of $\mathbb{Q}$. Different quadratic polynomials, for example $x^2+x+1$ and $x^2+$ $6 x+12$, can give rise to the same quadratic field $K$. Our next theorem gives a unique way of representing a quadratic field.

## 数学代写|数论作业代写number theory代考|Simple Extensions

Definition 5.5.1 (Simple extension) Let $K$ be a subfield of $\mathbb{C}$ and let $\alpha \in \mathbb{C}$. Let
$$K(\alpha)=\bigcap_{\substack{F \ \alpha \in F \ K \subseteq F \subseteq \mathbb{C}}} F,$$
where the intersection is taken over all subfields $F$ of $\mathbb{C}$, which contain both $K$ and $\alpha$. The intersection is nonempty as $\mathbb{C}$ itself is such a field. Since the intersection of subfields of $\mathbb{C}$ is again a subfield of $\mathbb{C}, K(\alpha)$ is the smallest field containing both $K$ and $\alpha$. We say that $K(\alpha)$ is formed from $K$ by adjoining a single element $\alpha$. A subfield $L$ of $\mathbb{C}$ for which there exists $\alpha \in \mathbb{C}$ such that $L=K(\alpha)$ is called a simple extension of $K$.
Clearly if $\alpha \in K$ then $K(\alpha)=K$.
For $K \subseteq \mathbb{C}$ and $\alpha \in \mathbb{C}$ let
$$L=\left{\frac{b_0+b_1 \alpha+\cdots+b_k \alpha^k}{c_0+c_1 \alpha+\cdots+c_h \alpha^h} \mid k, h \in \mathbb{N} \cup{0}, \begin{array}{l} b_0, \ldots, b_k, c_0, \ldots, c_h \in K, \ c_0+c_1 \alpha+\cdots+c_n \alpha^h \neq 0 \end{array}\right} .$$
Then $L$ is a subfield of $\mathbb{C}$ that contains both $K$ and $\alpha$. Moreover any subfield of $\mathbb{C}$ containing both $K$ and $\alpha$ must contain all the elements of $L$. Hence $L$ is the smallest subfield of $\mathbb{C}$ containing both $K$ and $\alpha$, so that $L=K(\alpha)$.

# 数论作业代写

## 数学代写|数论作业代写number theory代考|Algebraic Integers in a Quadratic Field

\begin{aligned} \mathbb{Q}(\alpha)= & \left{\frac{a_0+a_1 \alpha+\cdots+a_m \alpha^m}{b_0+b_1 \alpha+\cdots+b_n \alpha^n} \mid m, n\right. \text { (nonnegative integers), } \ & \left.a_0, \ldots, a_m, b_0, \ldots, b_n \in \mathbb{Q}, b_0+b_1 \alpha+\cdots+b_n \alpha^n \neq 0\right} . \end{aligned}

$$\mathbb{Q}(\alpha)=\left{\frac{e_0+e_1 \alpha}{f_0+f_1 \alpha} \mid e_0, e_1, f_0, f_1 \in \mathbb{Q},\left(f_0, f_1\right) \neq(0,0)\right} .$$

$$\frac{e_0+e_1 \alpha}{f_0+f_1 \alpha}=\left(\frac{e_0 f_0-a e_0 f_1+b e_1 f_1}{f_0^2-a f_0 f_1+b f_1^2}\right)+\left(\frac{e_1 f_0-e_0 f_1}{f_0^2-a f_0 f_1+b f_1^2}\right) \alpha,$$

$$\mathbb{Q}(\alpha)={x+y \alpha \mid x, y \in \mathbb{Q}},$$

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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。

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