Questions tagged [ideal-class-group]
The ideal-class-group tag has no usage guidance.
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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 ...
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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 ...
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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:...
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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.
...
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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 ...
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Invers of a fractional ideal in an order
i'm currently studying orders in quadratic number fields.
In detail, i study fields $\mathbb{Q}[\sqrt{-p}]$ and the order $\mathbb{Z}[\sqrt{-p}]$ with $p = 4l_1...l_r-1$ and $l_i$ small primes ...
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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 ...
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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 ...
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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 ...
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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{...
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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 $[I],[J]$ ...
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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 $N(I) = N(\mathfrak{p})$, ...
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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 ...
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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 $\mathbb{Z}\left[\frac{1+\sqrt{-39}}{2}\right]$)....
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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 ...
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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 ...
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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 ...
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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 ...
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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
$$
D=
\left\{ (u,v)\in\...
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What is the meaning of the ideal class group?
When I first learned about the ideal class group, I learned that it measures the failure of unique factorization in a number ring. The main justification for this is that a number ring has unique ...
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Is the class number of $K$ the number of factorizations an element of $\mathcal{O}_K$ can have?
Consider the number field $K = \mathbf{Q}(\sqrt{-5})$, which has ring of integers $\mathcal{O}_K = \mathbf{Z}[\sqrt{-5}]$. It is known that the class number of $K$ is $2$. It is also true that you can ...
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The class number of $\mathbb{Q}(\sqrt{-3},\sqrt{5})$
The following is my efforts on this question.
Let $K$ be the number field $\mathbb{Q}(\sqrt{-3},\sqrt{5})$, and $\mathcal{O}_K$ be the ring of integers of $K$. Let $k=-3\cdot5/(-3,5)^2=-15$. Since $-3\...
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Relation between $S$-ideal class group and usual ideal class group
For $\mathcal{O}_{K}$, the integer ring of a global field, we denote $S$ to be any set of primes of a global field $K.$ Let
$$\mathcal{O}_{K,S}:=\{x\in K\mid v_{\mathfrak{p}}\geq 0\text{ for }\...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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