# 数学代写|数论作业代写number theory代考|MATH3240

#### Doug I. Jones

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## 数学代写|数论作业代写number theory代考|Finiteness of the Ideal Class Group

In this section we use Minkowski’s linear forms theorem to show that every class in the ideal class group $H(K)$ of an algebraic number field $K$ contains an integral ideal of $O_K$ with norm less than a certain bound, called the Minkowski bound, that depends only on the degree of the field $K$ and the discriminant of $K$. For a particular algebraic number field $K$ these ideas give a method of determining the ideal class group $H(K)$.

Theorem 12.5.1 Let $K=\mathbb{Q}(\theta)$ be an algebraic number field of degree $n=r+2 s$, where $\theta$ has $r$ real conjugates and s pairs of nonreal complex conjugates. Let A be an integral or fractional ideal of $O_K$. Then there exists an element $\alpha(\neq 0) \in A$ such that
$$|N(\alpha)| \leq\left(\frac{2}{\pi}\right)^s N(A) \sqrt{|d(K)|} .$$
Proof: Let $\theta_1, \theta_2, \ldots, \theta_n$ be the conjugates of $\theta$. We reorder $\theta_1, \theta_2, \ldots, \theta_n$ in such a way that $\theta_1, \theta_2, \ldots, \theta_r \in \mathbb{R}$ and $\theta_{r+1}, \theta_{r+2}, \ldots, \theta_n \in \mathbb{C} \backslash \mathbb{R}$. As the complex conjugate of any conjugate of $\theta$ is also a conjugate of $\theta$ , we can further order $\theta_{r+1}, \theta_{r+2}, \ldots, \theta_n$ so that $\theta_{r+s+1}=\overline{\theta_{r+1}}, \ldots, \theta_n=\theta_{r+2 s}=\overline{\theta_{r+s}}$, where $r+2 s=n$. Let $\sigma_1, \ldots, \sigma_n$ be the $n$ monomorphisms : $K \rightarrow \mathbb{C}$ chosen so that $\sigma_i(\theta)=\theta_i$. Hence $\sigma_{r+s+t}=\overline{\sigma_{r+t}}(t=1, \ldots, s)$.

Let $\left{\alpha_1, \ldots, \alpha_n\right}$ be a basis for $A$. We define $n$ linear forms $L_j(\mathrm{x})(j=$ $1,2, \ldots, n)$ by
$$L_j(\mathrm{x})=\sum_{k=1}^n \sigma_j\left(\alpha_k\right) x_k .$$

## 数学代写|数论作业代写number theory代考|Algorithm to Determine the Ideal Class Group

The results of the previous section give us a method of determining all the ideal classes of a given algebraic number field $K$. To determine representatives of the ideal classes, we need only look at the integral ideals of $O_K$ with norm less than or equal to the Minkowski bound $M_K$. If $A$ is such an ideal then $N(P) \leq M_K$ for every prime ideal $P$ dividing $A$. Now $N(P)=p^f$ for some rational prime $p$ and some positive integer $f$ so the prime ideals occurring in the prime factorizations of the various integral ideals $A$ are all factors of rational primes $p \leq M_K$. Thus if we take each rational prime $p \leq M_K$, determine the prime ideal factorization of $\langle p\rangle$ in $O_K$, and form all possible products of the prime ideal factors of these various rational primes that yield ideals with norm $\leq M_K$ then we are sure to have at least one representative of every ideal class.

In particular, if every rational prime $\leq M_K$ factors into a product of prime ideals of $O_K$, each of which is a principal ideal, then $K$ has class number $h(K)=1$. For in this case every ideal of the type described here will also be principal.

Algorithm to find the ideal class group $H(K)$ of an algebraic number field $K$ :
Input. Algebraic number field $K=\mathbb{Q}(\theta)$.
Step 1. Determine $n=[K: \mathbb{Q}]$.
Step 2. Determine $r$ the number of real conjugates of $\theta$. Then $s=\frac{1}{2}(n-r)$.
Step 3. Determine $d(K)$.
Step 4. Compute the Minkowski bound $M_K=(2 / \pi)^s \sqrt{|d(K)|}$.
Step 5. Determine all rational primes $p \leq M_K$.
Step 6. Determine the prime ideal factorization of each principal ideal $\langle p\rangle$ in $O_K$ with $p$ as in Step 5.
Step 7. Determine all products of these prime ideals having norm $\leq M_K$.
Step 8. Determine the generators of $H(K)$ from the classes of these products.
Output. $H(K)$.
We illustrate this algorithm by finding the ideal class group of several algebraic number fields.

We denote the class containing the ideal $A$ by $[A]$ and the class of principal ideals by 1 .

# 数论作业代写

## 数学代写|数论作业代写number theory代考|Finiteness of the Ideal Class Group

$$|N(\alpha)| \leq\left(\frac{2}{\pi}\right)^s N(A) \sqrt{|d(K)|} .$$

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