Questions tagged [ideal-class-group]

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Calculating integral in order to prove Dirichlet’s class number formula for real quadratic number field

So, I'm trying to prove Dirichlet’s class number formula for a real quadratic number field. (Exc. 10.5.9 in Problems in Algebraic Number Theory” Murthy, Esmonde.) We define $$ D= \left\{ (u,v)\in\...
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4answers
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What is the meaning of the ideal class group?

When I first learned about the ideal class group, I learned that it measures the failure of unique factorization in a number ring. The main justification for this is that a number ring has unique ...
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Is the class number of $K$ the number of factorizations an element of $\mathcal{O}_K$ can have?

Consider the number field $K = \mathbf{Q}(\sqrt{-5})$, which has ring of integers $\mathcal{O}_K = \mathbf{Z}[\sqrt{-5}]$. It is known that the class number of $K$ is $2$. It is also true that you can ...
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The class number of $\mathbb{Q}(\sqrt{-3},\sqrt{5})$

The following is my efforts on this question. Let $K$ be the number field $\mathbb{Q}(\sqrt{-3},\sqrt{5})$, and $\mathcal{O}_K$ be the ring of integers of $K$. Let $k=-3\cdot5/(-3,5)^2=-15$. Since $-3\...
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1answer
117 views

Relation between $S$-ideal class group and usual ideal class group

For $\mathcal{O}_{K}$, the integer ring of a global field, we denote $S$ to be any set of primes of a global field $K.$ Let $$\mathcal{O}_{K,S}:=\{x\in K\mid v_{\mathfrak{p}}\geq 0\text{ for }\...
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25 views

Counting $S$-Integer Lattice Points

The fundamental principle underlying the geometry of numbers is that the number of lattice points in a round, compact subset $R \subset \mathbb{R}^n$ is roughly given by the volume of $R$. A very ...
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2answers
129 views

Why do we use *fractional* ideals in construction of the class group?

I am trying to give a simplified explanation of the class group to students at the upper undergraduate level, who likely have a basic understanding of ring and group theory. While my motivation for ...
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1answer
40 views

Dirichlet L-series inequality

Let $h$ be a class number of an imaginary quadratic number field of discriminant $d$. It holds that $h = k(d)\cdot L_d(1)$ where $k(d)$ is the Dirichlet structure constant and $L_d$ is the Dirichlet $...
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1answer
32 views

Primes Equidistribute in Ray Class Groups

Fix $K$ a number field and a modulus $\mathfrak m.$ Theory around the class number formula tells us that primes $\mathfrak p$ of $K$ equidistribute over all ideal classes in the normal class group, $...
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1answer
91 views

Finding the order of the ideal class group

This is an understanding question : to search for the order $n$ of the ideal class group, why do we only need to try to split the $(p)$ where p is a prime $\leq M$ ($M$ being the Minkowski bound) ? I ...
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50 views

How to show that an ideal is not principal in a generic number field

Suppose I'm trying to calculate the class group of $\mathbb{Q}(\zeta_{29})$, such as in this MathOverflow question. It is noted that Magma calculates the class group to be $(\mathbb{Z}/2\mathbb{Z})^3$...
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1answer
144 views

Checking if a prime ideal is principal for a particular cubic field

Consider the totally real cubic field $K = \mathbb{Q} (\alpha) = \mathbb{Q} [x]/(f)$ defined by $f = x^{3} - x^{2} - 9 x + 10$ (see https://www.lmfdb.org/NumberField/3.3.1957.1). The ring of integers ...
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Can anybody shed some light on the importance of the notion of compatibility of products of lattices in quaternion algebras?

I have started to study the theory of orders in quaternion algebras, but I am very new to the topic. In particular I am interested in the Class set associated with an order in such an algebra. I have ...
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1answer
203 views

Non-Linear Forms for all Prime Numbers [closed]

Edit to open the question: It looks like there are quadratic froms, satisfying these conditions. So, Is there any other form like quadratic form, for example, say cubic form or form of higher degree ...
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$\mathbb{A} \cap \mathbb{Q}[\alpha]$ is a principal ideal domain with $\alpha^3=\alpha+7$

I'm trying to solve exercise 20 of chapter 5 of the book Number Fields by Marcus. $$ \text{Prove that $\mathbb{A} \cap \mathbb{Q}[\alpha]$ is a principal ideal domain when $\alpha^3 = \alpha+ 7$} $$ I ...
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144 views

Class number of $\mathbb{Q}(\sqrt{-23})$ [duplicate]

I want to find the class number of $K=\mathbb{Q}(\sqrt{-23})$. First I found Minkowski bound by using $n=2$, $s=1$ and the $disc(K)=-23$. It is bigger than $3$. So, it is enough to check for prime $p=...
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1answer
449 views

Finding class number of quadratic number field using Minkowski bound

My understanding of this is as follows: In the general case, one has a quadratic number field $F$, which is always of the form $\mathbb{Q}(\sqrt{d})$ for some square-free integer $d$. Minkowski ...
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1answer
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Another Question Regarding Prime Ideals in $\mathbb{Q}[\sqrt{-199}]$

I have three prime ideals that "belong" to $\mathbb{Z}[\frac{1 + \sqrt{-199}}{2}]$: $P = (2, \frac{1 + \sqrt{-199}}{2})$ , $Q = (5, \frac{1 + \sqrt{-199}}{2})$, and $S = (7, 3 - \frac{1 + \sqrt{-199}...
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1answer
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Classifying Ideal Class Group

I wish to show that the ideal class group of the ring of integers of $\mathbb{Q}[\sqrt{-199}]$ is the cyclic group $\mathbb{Z}_9$ for some practice identifying ideal class groups. I know there is a ...
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Group Isomorphism Question

In this pdf (https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgroupKronecker.pdf) the author claims that the ideal class group of the ring of integers of $\mathbb{Q[\sqrt{-199}]}$ is the cyclic ...
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1answer
35 views

Condition that two ideal classes in the class group are equal

Let $\text{Cl}(K)$ denote the class group of a number field $K$. If $\frak{a}$ is a non-zero fractional ideal of $K$, write $\left[\frak{a}\right]$ for its class in $\text{Cl}(K)$. My lecture notes ...
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1answer
215 views

the narrow class number is a multiple of the class number

Let $K$ be a number field. $a \in K$ is said to be totally positive if $a^{\sigma}$ is positive for all real embeddings $\sigma$ of $K$. A principal ideal of $\mathcal{O}_K$ is said to be totally ...
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1answer
90 views

Prime Ideal, Ideal Norm

Consider $K=\mathbb{Q}(\sqrt{-11})$. It is quite straight forward to show that the Minkowski Bound $M_{K}<3$. It follows that every ideal class contains ideal of norm $\leq 2$. Now we show that $(...
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Question on Krull domains.

I'm stuck on a detail in Luther Claborn's paper Every Abelian Group is a Class Group. Recall that we say an integral domain $A$ is a Krull domain if \begin{align*} &\text{(1) }A = \bigcap_{P \in ...
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46 views

Using the Minkowski bound to prove that a ring is principal

Let $\alpha =e^{2i\pi/7}$. Prove that $\Bbb Z[\alpha ]$ is principal. I try to prove that the class group of $\Bbb K:=\Bbb Q[\alpha ]$ is trivial which I think is sufficient condition to prove ...
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What is the Chern class of a line bundle over a number ring?

Question: Let $F$ be a finite extension over $\def\q{\mathbb Q}\q$. Let $\mathcal O_F$ be the integral closure of $\mathbb Z$ in $F$. Then if I am not mistaken, a line bundle (an invertible sheaf) ...
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Computing Class Group of $\mathbb{R}[x, y] / \langle x^2+y^2 -1 \rangle$ [duplicate]

$ \mathbb{R}[x, y] / \langle x^2+y^2 -1 \rangle$ is a Dedekind domain. How to compute its ideal class group? Any idea.
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Class group - product of proper ideals

I am looking at the definition (Sutherland, last page) of class group as the set of proper ideals of $\mathcal{O}$ modulo homothety, where: $\mathcal{O}$ is an order in an imaginary quadratic number ...
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1answer
182 views

Class group and localization in number fields

In The Rising Sea, Vakil says that $A = \mathbb{Z}[\sqrt{-5}]$ shows that the property "being the spectrum of a UFD" is not an affine-local property. Concretely, he points out that $D(2) = \...
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1answer
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class group calculation for $\Bbb Q(i)$

I am trying to understand this computation of the class group of $\Bbb Q(i)$. I don't understand why $2\mathcal O_K=(1+i)^2$. I don't know how to calculate $\mathcal O_K$, I know it's hard in general....
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1answer
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If $A$ is a fractional ideal of $\mathbb{Q}(\sqrt{-m})$, then $A^{1+\sigma}$ is an ideal in $\mathbb{Q}$.

I'm stuck on this little detail in Washington's intro to cyclotomic fields. Let $M = \mathbb{Q}(\sqrt{-m})$, and $A$ be a fractional ideal of $M$. With $Gal(M/\mathbb{Q})= \{1,\sigma\}$, Washington ...
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Non-unique factorization of ideals in $\mathbb{Z}[t,t^{-1}]$

Edited version: In a Dedekind domain $R$, every nonzero proper ideal factors uniquely as a product of prime ideals. If $R$ is a Noetherian domain, then by this post any ideal $I$ which does factor ...
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1answer
153 views

class number of groups and rings

The class number of a group is defined by the number of conjugacy classes. The class number of a ring(specifically a number field or its algebraic integers) is defined by the order of ideal class ...
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1answer
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Mordell curve when $d\equiv 3 \pmod 4$

One way to attack, say, $y^2 + 65 = x^3$ in integers is to factor as $(y+\sqrt{-65})(y-\sqrt{-65})= x^3$, show that the ideals $(y+\sqrt{-65})$, $(y-\sqrt{-65})$ are coprime - hence both cubes of an ...
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1answer
180 views

Quadratic field ideal find $\mathbb{Z}$-basis given a $\mathcal{O}_K$-basis

Suppose we are working in an imaginary quadratic number field $\mathbb{Q}(\sqrt{d})$ (so $d$ is a fundamental discriminant with $d < 0$). Now in the ring of integers $\mathcal{O}_K$ suppose we ...
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1answer
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On the class group of $\mathbb{Q}(\sqrt{-199})$

The Minkowski bound for $\mathbb{Q}(\sqrt{-199})$ (with ring of integers $\mathbb{Z}(\frac{1+\sqrt{-199}}{2})$) is $\frac{2}{\pi}\sqrt{199} < 11$, so we need to consider $2$, $3$, $5$, $7$. Since $|...
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1answer
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Finding a prime ideal whose class is order 10 in the ideal class group

"By factorising the ideal $(4+\sqrt{-74})_{R}$, or otherwise, find a prime ideal whose class $[P]$ in the ideal class group $Cl(R)$ has order 10" $R = \mathbb{Z}[\sqrt{-74}]$ So I've managed to get ...
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Determine the generator of an ideal of ring of integers

I am trying to find the generators of the ideal $(3)$ in the ring of integers of $\mathbb{Q}[\sqrt{-83}]$ the ring of integers is $\mathbb{Z}\left[\frac{1+\sqrt{-83}}{2}\right]$ I evaluated the ...
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1answer
127 views

Does the abelianization of the Galois group determine the ideal class group?

Let $K$ be an algebraic number field, assumed to be Galois, with Galois group $G = Gal(K/\mathbb{Q})$. Is knowing the abelianization of $G$ alone, without other information on $K$, enough to ...
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202 views

On the class group of $\Bbb Q(\sqrt{-d})$, $d = n^g - 1$

Let $g > 1$ and $n \geq 3$ be integers such that $n$ is odd and $d = n^g - 1$ is squarefree. Prove that the class group of $\Bbb Q(\sqrt{-d})$ contains an element of order $g$. Here is my attempt: ...
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1answer
92 views

Class group of contains an element of order greater than $n$

Let $p$ be a prime such that $p \equiv 5 \pmod {12}$ and let $n$ be a positive integer such that $p > 3^n$. Prove that the ideal class group of $\mathbb{Q}(\sqrt{-p})$ contains an element of order ...
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Structure of class group of quadratic forms negative discriminant

I am trying to work out the structure of the class groups for discriminants of some simple forms. I am interested in the cases where the discriminant is a prime number and where it is of the form $d =...
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What is the ideal class group? [duplicate]

I'm trying to understand imaginary quadratic fields and the ideal class group seems to be really important, yet I cannot find a simple explanation of it and why it's important. I understand that it is ...
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1answer
57 views

Help with the derivation of the class number formula

I'm interested in the class number formula derived from the Dedekind zeta function, but I have no idea how to derive it. From what I've read, turning the Dedekind zeta function into a normal Dirichlet ...
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1answer
187 views

Neukirch's interpretation of unit group and class group

In page 22 from Neukirch's Algebraic Number Theory, he defines the class group $Cl_K$ of a number field $K$ to be the quotient of group of fractional ideals $J_K$ by the subgroup of principal ideals $...
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1answer
164 views

Good way to check if an ideal is principal in cubic extensions

I've recently been tacking the computation of class groups. I've noticed that when dealing with a ramified prime ideal in a quadratic field $\mathbb{Q}(\sqrt{d}), d \in \mathbb{Z}$ squarefree, it's ...
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1answer
121 views

Contradictory results when computing the Ideal class group of $\mathbb{Q}(\sqrt{-7})$

I seem to have arrived at some contradictory results in computing this group, would you mind helping me resolve this? By Sage + internet I find that it should be true that this class group is trivial....
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What is the relation between torsion elements of the class group and covering spaces of curves?

For a Dedekind domain $A$ we have the following result relating torsion elements of the class group to (mostly) unramified extensions: If $a\in A$ is such that there exists an ideal $I$ with $I^n=(a)$...
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42 views

How to guarantee non existence of order 4 elements in class group of a maximal order

If we know the prime factorisation of the fundamental negative discriminant $\Delta_K$, say $n$ prime factors, then we are guaranteed that $2^{n-1}\mid h_K$, the class number of $C(\mathcal{O}_K)$ ...
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181 views

Understanding $p$-rank, $p$-Sylow subgroup, $2$-part, odd and even part of a group.

As the title says, I would like to be clear about what each of those terms mean in an ideal class group setting. I would like someone to correct me where I am wrong. The reason for this clarification ...