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

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. ...
Robin's user avatar
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Calculation of the Picard group and the class group [closed]

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

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 $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ ...
Pont's user avatar
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1 answer
117 views

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

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}(...
Xiong Jiangnan's user avatar
2 votes
0 answers
58 views

Invertible lattices

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[\...
Mystery girl's user avatar
2 votes
1 answer
96 views

What information does the numerical norm provide (and why)?

For an algebraic number field $K$ with ring of integers $𝒪_K$, the numerical norm $ℕ(𝔞)$ of an ideal $𝔞⊆𝒪_K$ is defined to be the (finite) index of abelian groups $[𝒪_K:𝔞]$, or equivalently $|𝒪...
Elizabeth Henning's user avatar
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The plane in the quadric 3-fold is not a (set-theoretic) hypersurface

I think my question has a top-bottom answer, but as of yet I am not familiar enough with divisors and class groups to be sure of what I am claiming. I also include an "elementary answer" to ...
Andrei.B's user avatar
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1 answer
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Ideal class groups of number fields

Let $K\subseteq L$ be a two number fields with ring of integers $\mathcal O_K$ and $\mathcal O_L$ respectively. And let $I(K)$ and $I(L)$ (resp. $P(K)$ and $P(L)$) the fractional ideals (resp. the ...
Luis Antonio Sanchez's user avatar
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Neukirch 'Algebraic number theory', p73, Prop 11.6. there is a surjection $Cl(O) \to Cl(O(X))$.

This is a question related to Neukirch 'Algebraic number theory', $p 71$, $Prop 11.6$ (https://web.math.ucsb.edu/~agboola/teaching/2021/fall/225A/neukirch.pdf). Let $o$ be a Dedekind domain and let $o(...
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An exact sequence $1 \to {O_{K,S}}^{\times}/{{O_{K,S}}^{\times}}^2 \to K(S,2) \to Cl(K,S)[2] \to 1$

Let $K$ be an imaginary number field.Let $S$ be finite set of places of $K$. Let $K(S,2)\stackrel{\mathrm{def}}{=} \{b\in K^{\times}/{K^{\times}}^2 \mid v(b)≡0\mod2, \forall v\notin S \}$ Let $S-$ ...
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Concrete example of S-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 }\...
Pont's user avatar
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Totally ramified principal ideal

Let $K$ be a number field of degree $d=[K,\mathbb{Q}]$ and suppose that the class number $h_K$ of $K$ is comprime with $d$. Let $p$ a prime of $\mathbb{Q}$ and assume that $(p)=\mathcal{P}^d$ is ...
Mario's user avatar
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Find a non-trivial element in the class group of $\mathbb{Q}( \sqrt{−5})$.

a. Find a non-trivial element in the class group of $\mathbb{Q}(\sqrt{−5})$. b. Show that the class group of $\mathbb{Q}( \sqrt{−5})$ has order two. For part a: I know that the class group is the ...
user1052623's user avatar
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Hint for an Imaginary quadratic field question

I am kinda stuck and am looking for a hint on how to proceed. Let $m > 2$ be an integer such that $p = 4m − 1$ is prime. Suppose that the ideal class group of $L = Q(\sqrt{−p})$ is trivial. Show ...
RickSmith's user avatar
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Class group of quadratic field extension $\mathbb Q(\sqrt{-69})$ has order 8

I'm practicing finding class groups, in this case for $K = \mathbb{Q}(\sqrt{-69})$, and found the class number $h_K$ to be 16, with $C_K \cong C_2 \times C_2 \times C_4$ (cyclic product) whereas https:...
George's user avatar
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Showing that $\mathbf{Z}$ adjoin a primitive seventh root of unity is a UFD.

What I have done so far is first note this is the ring of integers over the seventh cyclotomic field. This has degree 6 and only complex embeddings. Then I computed minkowskis bound to be around 4.4. ...
RickSmith's user avatar
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7 votes
1 answer
124 views

Parity of the class number of cyclotomic fields

I am interested if there are any heuristics on the parity of the class number of $\mathbb{Q}(\zeta_p)$ where $\zeta_p$ is a primitive root of unity. Is it true that it is odd infinitely many often? Is ...
did's user avatar
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1 answer
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Janusz Algebraic Number Fields, Exercise 4, Page 19

I am currently working through Algebraic Number Fields by Janusz and towards the bottom of page 19 he leaves an exercise to the reader. The exercises states Let $S$ be a multiplicative set in the ...
badatalg's user avatar
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Question on the proof that if exactly on prime ramifies in a $p$-extension $L/K$, then $p \mid h_L$ implies $p \mid h_K$.

Let $L/K$ be a Galois extension of number fields with $p$-power degree ($p$ prime) in which exactly one prime, say $\mathfrak{q}$ ramifies. Let $H_L$ be the $p$-Hilbert class field for $L$ and ...
matt stokes's user avatar
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Ideal class group of $ \Bbb{Z}[ \sqrt{-5}]$

Ideal class group of $ \Bbb{Z}[ \sqrt{-5}]$ is known to be {$(1),I=(2,1+\sqrt{-5}$}. This is a group of order $2$, so $I^2$ must be $(1)$. But after calculating,$I^2=(2)$. $(1)$ is not the same ideal ...
Pont's user avatar
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Brumer theorem about ideal class group

I am reading chapter 9 in Cassels-Frohlich Algebraic Number Theory, this chapter is about class field tower. In this chapter the following claim is due to Brumer: There exist a function $c(n)$ \begin{...
re'em waxman's user avatar
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Is there a prime decomposing into order $r$ ideals for every every prime factor $r$ of $h(K)$ in a quadratic number field $K$?

Let $K$ be a quadratic number field, $h(K)$ its class number. Is it true that: for every prime factor $r$ of $h(K)$, there is a prime $p$ that decomposes in $K$, and, if $(p) = IJ$, then $[I],[J]$ ...
Jianing Song's user avatar
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1 vote
1 answer
132 views

Ideals having the same norm as a prime ideal in ring of integers of a number field

Let $K$ be a number field and $\mathcal{O}_K$ its ring of integers. Let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_K$, and $I$ be an ideal of $\mathcal{O}_K$ such that $N(I) = N(\mathfrak{p})$, ...
Jianing Song's user avatar
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A question on Ray class groups and the relative degree of a prime

I have been stuck on a detail in the proof of Theorem 4 in R. Gold, The non-triviality of certain $\mathbb{Z}_l$-extensions (see also the the lemma and the following statement on page 3). Here is the ...
matt stokes's user avatar
1 vote
1 answer
192 views

Diophantine equation with class group techniques

Solve in integers $x^2 - x + 10 = y^3$. (You may use the fact that the class number of $\mathbb{Q}(\sqrt{-39})$ is $4$ and that its ring of integers is $\mathbb{Z}\left[\frac{1+\sqrt{-39}}{2}\right]$)....
DesmondMiles's user avatar
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Basic questions on Picard group (references)

I want a reference/explanation for things as follows. For nice enough commutative ring $A$, we have $Pic(A[[x]]) \cong Pic(A)$. Also, I want to know about comparing $Pic(A((x)))$ and $Pic(A)$, for ...
Sasha's user avatar
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2 votes
0 answers
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Divisibility of class number of infinitely many real quadratic fields by a given number $g.$

I want to prove the following result. If $d$ is a square free integer of the form $ d=n^{2g}+1$,where $ n>4$,then $\ g|h$,where $ h$ is the class number of the field $ \mathbb Q(\sqrt{d})$. There ...
Soumyadip Sarkar's user avatar
1 vote
0 answers
114 views

Ring of integers of cubic number fields

I am trying to give an explicit expression of some ideal class groups of cubic number fields. I taking as a reference for the examples Number Fields by Daniel A. Marcus, in particular the cubic field ...
00strich's user avatar
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1 answer
149 views

Elementary Proof that the class number of $\mathbb{Z}[\sqrt{-5}]$ is $2$.

I am asking for clarification on the answer of Hagen von Eitzen to the question "Prove that the class number of $\mathbb{Z}[\sqrt{-5}]$ is $2$. He claims that given any non-principal ideal $J$ in ...
Sven-Ole Behrend's user avatar
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0 answers
57 views

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\...
slowpoke's user avatar
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27 votes
5 answers
3k views

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 ...
Adithya Chakravarthy's user avatar
6 votes
4 answers
261 views

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 ...
Adithya Chakravarthy's user avatar
2 votes
0 answers
90 views

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\...
L. Yhui's user avatar
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3 votes
1 answer
503 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 }\...
Pont's user avatar
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1 vote
0 answers
108 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 ...
Ashvin Swaminathan's user avatar
5 votes
2 answers
330 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 ...
Ben Doyle's user avatar
1 vote
1 answer
152 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 $...
sugyman's user avatar
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1 vote
1 answer
74 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, $...
Derive Foiler's user avatar
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1 answer
375 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 ...
hyuno's user avatar
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2 votes
0 answers
74 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 $(\mathbb{Z}/2\mathbb{Z})^3$...
Bob Jones's user avatar
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3 votes
1 answer
192 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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0 answers
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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 ...
Giovanni Deligios's user avatar
2 votes
1 answer
247 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 ...
Consider Non-Trivial Cases's user avatar
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0 answers
106 views

$\mathbb{A} \cap \mathbb{Q}[\alpha]$ 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 ...
Sant97's user avatar
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0 answers
332 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=...
Tuğba Yesin's user avatar
3 votes
1 answer
1k 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 ...
user829347's user avatar
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1 vote
1 answer
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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}...
Debbie's user avatar
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4 votes
1 answer
244 views

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 ...
Alice Jennings's user avatar
5 votes
1 answer
121 views

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 ...
Debbie's user avatar
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1 vote
1 answer
86 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 ...
Fortox's user avatar
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