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# 数学代写|数论代写Number Theory代考|MATH11226 Class Group and Class Number

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## 数学代写|数论代写Number Theory代考|Finiteness of Class Number

Definition. Let $K$ be an algebraic number field. Let $G(K)$ denote the group of all non-zero fractional ideals of $K$ and $P(K)$ the subgroup of all non-zero principal fractional ideals of $K$. The group $G(K) / P(K)$ is called the class group or the ideal class group of $K$ and its order is called the class number of $K$. It will be shown that $G(K) / P(K)$ is a finite group. A member of $G(K) / P(K)$ is called an ideal class of $K$.
In this section, we prove.

Theorem 8.1 The group of ideal classes of an algebraic number field is finite.
We shall first prove the following theorem from which the above theorem will be deduced.

Theorem 8.2 Let $K$ be an algebraic number field different from $\mathbb{Q}$. Then in every ideal class of $K$, there exists an integral ideal B of $\mathcal{O}K$ such that $N(B)<\sqrt{\left|d_K\right|}$. Proof Let $\mathcal{C}$ be any ideal class of $K$ and $C$ be a fixed ideal belonging to $\mathcal{C}$. Since $C^{-1}$ is a fractional ideal, there exists a non-zero element $\beta$ of $\mathcal{O}_K$ such that $\beta C^{-1} \subseteq \mathcal{O}_K$. Then $A=\beta C^{-1}$ is an integral ideal of $\mathcal{O}_K$ such that $A C$ is a principal ideal. Fix a $\mathbb{Z}$-basis $\left{\alpha_1, \ldots, \alpha_n\right}$ of $A$. Let $\sigma_1, \ldots, \sigma{r_1}$ be all the real isomorphisms of $K$ and $\sigma_{r_1+1}, \ldots, \sigma_{r_1+2 r_2}$ be the non-real isomorphisms of $K$ which are arranged such that $\bar{\sigma}{r_1+j}=\sigma{r_1+r_2+j}$ for $1 \leq j \leq r_2$.
Define linear forms $L_1, L_2, \ldots, L_n$ by
$$L_i\left(x_1, x_2, \ldots, x_n\right)=\sigma_i\left(\alpha_1\right) x_1+\sigma_i\left(\alpha_2\right) x_2+\cdots+\sigma_i\left(\alpha_n\right) x_n .$$
The absolute value of the determinant of these linear forms is
$$\left|\operatorname{det}\left(\sigma_i\left(\alpha_j\right)\right){i, j}\right|=\sqrt{\left|D{K / \mathbb{Q}}\left(\alpha_1, \ldots, \alpha_n\right)\right|}$$
which in view of Theorem $2.15$ equals $\left[\mathcal{O}_K: A\right] \sqrt{\left|d_K\right|}$. Define constants $c_i$ by
$$c_i=\left(\left[\mathcal{O}_K: A\right] \sqrt{\left|d_K\right|}\right)^{\frac{1}{n}} \text { for } 1 \leq i \leq n .$$

## 数学代写|数论代写Number Theory代考|Minkowski’s Convex Body Theorem

Definition. A set $S$ contained in $\mathbb{R}^n$ is said to be convex if whenever $x, y \in S$, then $\lambda x+(1-\lambda) y \in S$ for all $\lambda \in \mathbb{R}$ such that $0 \leq \lambda \leq 1$.

Definition. A set $S$ contained in $\mathbb{R}^n$ is said to be centrally symmetric if whenever $x \in S$, then $-x \in S$.

Example 8.8 Let $A=\left(a_{i j}\right){n \times n}$ be a non-singular matrix with entries from $\mathbb{R}$ and $c_1, c_2, \ldots, c_n$ be fixed positive constants. We show that the set $$P=\left{x \in \mathbb{R}^n:\left|a{11} x_1+a_{12} x_2+\cdots+a_{1 n} x_n\right|<c_1, \ldots,\left|a_{n 1} x_1+\cdots+a_{n n} x_n\right|<c_n\right}$$
is a convex set. To verify this, define linear forms $L_1, L_2, \ldots, L_n$ in $X=$ $\left(X_1, X_2, \ldots, X_n\right)$ by $L_i(X)=\sum_{j=1}^n a_{i j} X_j, 1 \leq i \leq n$. Suppose $x, y \in P$ and $\lambda \in \mathbb{R}$ is such that $0 \leq \lambda \leq 1$. We have $\left|L_i(x)\right|<c_i$ and $\left|L_i(y)\right|<c_i$ for $1 \leq i \leq n$. Now $\left|L_i(\lambda x+(1-\lambda) y)\right|=\left|\lambda L_i(x)+(1-\lambda) L_i(y)\right| \leq \lambda\left|L_i(x)\right|+(1-\lambda)\left|L_i(y)\right|<c_i$ for all $i$. Thus $\lambda x+(1-\lambda) y \in P$.

Definition. By an $n$-dimensional lattice in $\mathbb{R}^n$, we mean a subgroup of $\mathbb{R}^n$ which is generated as a group by $n$ linearly independent vectors over $\mathbb{R}$. Such a set of generators is called a basis of the lattice. If $\left{A_1, A_2, \ldots, A_n\right}$ and $\left{B_1, B_2, \ldots, B_n\right}$ are two bases of a lattice $\mathfrak{L}$, then there exists an $n \times n$ unimodular matrix $U$ such that
$$\left[\begin{array}{c} A_1 \ A_2 \ \vdots \ A_n \end{array}\right]=U\left[\begin{array}{c} B_1 \ B_2 \ \vdots \ B_n \end{array}\right]$$
So the absolute value of the determinant of the matrix with row vectors $A_1, A_2, \ldots, A_n$ is well defined. It is called the determinant of $\mathfrak{L}$. In what follows all lattices under consideration are $n$-dimensional.

For example, $\mathfrak{L}_0=\left{\left(a_1, \ldots, a_n\right): a_i \in \mathbb{Z}\right}$ is a lattice in $\mathbb{R}^n$ called the fundamental lattice or integral lattice and has got basis ${(1,0, \ldots, 0),(0,1,0, \ldots, 0), \ldots$, $(0,0, \ldots, 1)}$. Clearly $\operatorname{det}\left(\mathfrak{L}_0\right)=1$.

## 数学代写数论代写Number Theory代考|Finiteness of Class Number

$G(K) / P(K)$ 称为类群或理想类群 $K$ 它的顺序称为类号 $K$. 这将表明 $G(K) / P(K)$ 是一个有限群。的成员 $G(K) / P(K)$ 被称为

$$L_i\left(x_1, x_2, \ldots, x_n\right)=\sigma_i\left(\alpha_1\right) x_1+\sigma_i\left(\alpha_2\right) x_2+\cdots+\sigma_i\left(\alpha_n\right) x_n .$$

$$\left|\operatorname{det}\left(\sigma_i\left(\alpha_j\right)\right) i, j\right|=\sqrt{\left|D K / \mathbb{Q}\left(\alpha_1, \ldots, \alpha_n\right)\right|}$$
㧛于定理 $2.15$ 等于 $\left[\mathcal{O}K: A\right] \sqrt{\left|d_K\right|}$. 定义常量 $c_i$ 经过 $$c_i=\left(\left[\mathcal{O}_K: A\right] \sqrt{\left|d_K\right|}\right)^{\frac{1}{n}} \text { for } 1 \leq i \leq n .$$

## 数学代写|数论代写Number Theory代考|Minkowski’s Convex Body Theorem

〈left 的分隔符缺失或无法识别

$L_i(X)=\sum_{j=1}^n a_{i j} X_j, 1 \leq i \leq n$. 认为 $x, y \in P$ 和 $\lambda \in \mathbb{R}$ 是这样的 $0 \leq \lambda \leq 1$. 我们有 $\left|L_i(x)\right|<c_i$ 和 $\left|L_i(y)\right|<c_i$ 为了
$1 \leq i \leq n$. 现在 $\left|L_i(\lambda x+(1-\lambda) y)\right|=\left|\lambda L_i(x)+(1-\lambda) L_i(y)\right| \leq \lambda\left|L_i(x)\right|+(1-\lambda)\left|L_i(y)\right|<c_i$ 对所有人 $i$. 因此 $\lambda x+(1-\lambda) y \in P$

〈left 的分隔符缺失或无法识别 和〈left 的分隔符缺失或无法识别 是格子的两个基 $\mathfrak{L}$, 那么存 在一个 $n \times n$ 单模矩阵 $U$ 这样
$$\left[A_1 A_2 \vdots A_n\right]=U\left[B_1 B_2 \vdots B_n\right]$$

## MATLAB代写

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