# Questions tagged [ideal-class-group]

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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 ...
1 vote
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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 ...
1 vote
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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 ...
30 views

### 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 ...
1 vote
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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 ...
1 vote
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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 $[I],[J]$ ...
1 vote
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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 $N(I) = N(\mathfrak{p})$, ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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1 vote
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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 ...
1 vote
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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 ... 1 vote
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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 ...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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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1 vote
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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....
1 vote
### 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 ...
### 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 ...