Questions tagged [ideal-class-group]

Question 1

The Idoneal numbers have the property that if $d$ is an Idoneal number, then there is a congruence relation on primes $p$ such that $p$ satisfying the congruence relation iff $p = x^2 + dy^2$ has a ...

Question 2

The Minkowski theorem: Let $I$ be a fractional ideal of $O_K$, then there exists an integral ideal $J$ with $N(J)\leq\frac{n!}{n^n}(\frac4\pi)^s\sqrt{\Delta_K}$ s.t. $I$ and $J$ lie in the same ideal ...

Question 3

I have a problem when reading a paper "Mildly Short Vectors in Cyclotomic Ideal Lattices in Quantum Polynomial Time" about algebraic lattice theory. $K$ is cyclotomic field $\mathbb{Q}(\...

Question 4

Let $\mathbb{Q}(\sqrt{d})$ be finite extension with square free $d$, with $R_d$ being integral closure of subring $\mathbb{Z}$. It is known that $R_d=\mathbb{Z}[\sqrt{d}]$ if $d\not\equiv 1\pmod{4}$ ...

Question 5

Given the ring of integers created by adding the square root of $-d$ to $\mathbb{Z}$ for some positive integer $d$, is there a characterization of $d$ for when the class group of that ring has a size ...

Question 6

Let $K$ be an imaginary quadratic number field and $\mathrm{Cl}_K$ be the ideal class group of $K$. The main theorem of what we call genus theory is: $\#{\mathrm{Cl}}_K[2] = 2^{r-1}$ where $r$ is the ...

Question 7

$\textbf{Calculate the ideal class group of $K=\mathbb{Q}(\sqrt[3]{11})$}$: Let $\alpha=\sqrt[3]{11}.$ We need the fact that the ring of integer of $K$ is $\mathbb{Z}[\alpha]$. One basis:$\{x_1,x_2,...

Question 8

Let $K=\mathbb{Q}(\sqrt {-5})$. We have shown that $\mathcal{O}_K$ has the integral basis $1,\sqrt{-5}$ and $D=4d=-20$. By computing the Minkowski's constant:$$M_K=\sqrt{|D|}\Big(\frac{4}{\pi}\Big)^{...

Question 9

In the answer to the Motivation behind the definition of ideal class group, I have seen one by user Alex Youcis and he/she claimed that the following is a short exat sequence: $$1\to \mathcal{O}_k\to \...

Question 10

I'm trying to compute the ideal class group of $\mathbb Q(\sqrt{2},\sqrt{3})$, and I would like to know if my calculations are right and if I could improve my arguments. Let $K=\mathbb Q(\sqrt{2},\...

Question 11

I'm learning about the ideal class group of a number field, and am trying a few exercises where I calculate $\mathbb{Q}(\sqrt{d})$ for $d \in \mathbb{Z}$ for various $d$. I'd like to check my work. ...

Question 12

I am thinking about the computation of the class group and the Picard group for the case of Number fields over $\mathbb{Q}$ and $\mathbb{F}_p(t)$ Complex varieties I would like to know what kinds of ...

Question 13

Let $p$ be a negative prime number such that $p \equiv 5\pmod 8$. Let $K = \mathbb{Q}(\sqrt{p})$ and denote its ideal class group by $Cl_K$. I aim to prove that $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ ...

Question 14

I want to calculate the ideal class group of $K=\mathbb{Q}(\sqrt[3]{5})$ The ring of integers $O_K=\mathbb{Z}[\sqrt[3]{5}]$, discriminant $d_K=-27\cdot5^2=-675$, Minkowski constant $M_K=\frac{3!}{3^3}(...

Question 15

I am intersted in Exercise 7 from Chapter 5 of this notes: "Let $V$ be the set of non-zero lattices $L \subset \mathbf{C}$ that satisfy $x \cdot L \subset L$ for every $x \in \mathbf{Z}\left[\...

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Questions tagged [ideal-class-group]

What is the largest integer $d$ such there is a congruence relation on primes p so that $x^2 + dy^2 = p^2$ has a non-zero integral solution?

Connection between Minkowski theorem and adelic Minkowski theorem

Norm map from cyclotomic field to its maximal real subfield

Elementary Method for Calculating Ideal Class Group

Imaginary quadratic fields with class group of even order

Reference for the main theorem of genus theory of ideal class group

Calculate the ideal class group of $K=\mathbb{Q}(\sqrt[3]{11})$

Find the ideal class group of $\mathbb{Q}(\sqrt{-5})$ by using the factorization theorem

Short exact sequence in the ideal class group

Ideal Class Group of $\mathbb{Q}(\sqrt{2}+\sqrt{3})$

(Reference request) List of (isomorphism classes of) ideal class groups of $\mathbb{Q}(\sqrt{d})$.

Calculation of the Picard group and the class group [closed]

There exists imaginary quadratic extension which trivialized 2-part of ideal class group

Ideal class group of $\mathbb{Q}(\sqrt[3]{5})$

Invertible lattices

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