Questions tagged [ideal-class-group]

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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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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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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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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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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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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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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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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 ...