Questions tagged [ideal-class-group]

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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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51 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
36 views

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
54 views

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
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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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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
57 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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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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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
53 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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53 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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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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Form Class group - Special cases

I am trying to find special cases for when the form class group will have a predictable structure. I am specifically interested in the case of prime discriminants or relating the structure for non-...
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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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Deterministic algorithm for class group of Binary quadratic forms

I want to find the class group of a given negative discriminant. I know of Shanks method but this is not deterministic. I can of course brute force the problem by finding the class group and then ...
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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
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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
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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
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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
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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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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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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 ...
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weaker upper bound of class number of O_k

can anybody explain how the proof of this corollary is concluded? i understood the proof right before the underlined sentence but still do not get how the argument before that leads to the underlined ...
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Calculating class numbers

It is a general question about simple examples of calculating class numbers in quadratic fields. Here are an excerpt from Frazer Jarvis' book Algebraic Number Theory: "Example 7.20 For $K=\mathbb{Q}(\...
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161 views

Finding units in the ring of integers

In these notes about number theory, trying to compute the class group of the number field $K=\mathbb{Q}[X]/(g(X))$ where $g(X)=X^3+X^2+5X-16$, the author manages to find a fundamental unit by ...
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Calculating this class number

Let $f = x^5+2x^4-2$ and $\alpha \in \mathbb C$ with $f(\alpha) = 0$. Show that $\mathbb Z [\alpha ]$ is a principal ideal ring. What I have done so far: My idea was to first prove that $\mathcal ...
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Ideal Class Group Calculation: How to conclude the classes of two ideals are distinct

I am frequently attempting to compute class groups, with a pretty standard approach: Calculate the Minkowski bound, and list the primes less than this bound. Factor $(p)$ into prime ideals (usually ...
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Confusion about the class group

Let $K$ be a quadratic number field, assume that the related Minkowski bound is $10$. We know that the class group $Cl(\mathcal{O}_K)$ is generated by the classes of prime ideals $[\mathfrak{p}]$ ...
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How to compute quotient rings?

I am discovering class numbers and ideal class groups. Going through Wikipedia, the example $\mathbb{Z}[\sqrt{-5}]$ is considered along with the ideal $J=(1+\sqrt{-5}, 2)$. Two facts are troubling me (...
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1answer
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Fact about sum of ideal classes in the ideal class group

Let $K$ be a number field with ring of integers $\cal{O}_K$ and class group $\cal{O}_K$. Let $\mathfrak{p},\mathfrak{q}$ be two integral ideals such that $\mathfrak{p}\mathfrak{q} = (\alpha)$ for ...
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Prove a certain imaginary quadratic field contains three ideal classes.

I'm trying to show $\mathbb{Z}[\sqrt{-31}]$ has three distinct ideal classes, but something keeps going wrong and I can only find the identity and one other class. I found $|d_{K}| = 31$ because $-31 ...
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Example of principal ideals in the Class Group

I am working on the analytic class number formula. There is a step that I want to understand and need some help. Here is the setup: Let $K$ be a number field with class group $Cl_K$, $C$ be a class. ...
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Is the ratio of the norms of generators in an ideal well defined?

Let $K$ be a number field with ring of integers $\mathcal{O}$ containing a non-principal ideal $\mathfrak{i}$. Let us find generators $(a_1 ... a_n)$ for $\mathfrak{i}$ such that $N(\prod_{i=1}^na_i)$...
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divisibility of class numbers of composite two abelian number fields

Let $p$ be a regular prime i.e. $p\nmid h_p$, ($h_p$ denotes the class number of $\mathbb{Q}(\mu_p)$). Let $F$ be an abelian number field such that $p\nmid h_F$ and $p\nmid [F:\mathbb{Q}]$. Do we ...
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Conditions for when a class is a square in the ideal class group of a quadratic field.

Let Q($\sqrt{d}$) be a number field with discriminant $D$ such that $d \equiv 1,3$ (mod 4). Let $\mathcal{O}$ be its ring of integers and $cl(\mathcal{O})$ the respective class group. Consider an ...
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Reference request - Lattices for studying number fields

I am working on number fields and I have just started studying ideal class group. I came across with the Minkowski's convex body theorem, which simply says: $$ \text{Let } L \text{ be a lattice in } \...
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Lower bound for cardinality of class group

Let $p\equiv 2\pmod 3$ be a prime number such that $p>3^m$ for some $m\in\mathbb N$. Prove that the class number of $K=\mathbb{Q}(\sqrt{-p})$ is bounded below by $m$. I thought using Minkowski ...
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Is there an 'easy' way to calculate $K_0(\mathbb{Z}[C_p])$?

For $C_2$ the cyclic group of order 2, I want to calculate $\tilde{K}_{0}(\mathbb{Z}[C_2])$. Now so far, I know by a theorem of Rim that $\tilde{K}_{0}(\mathbb{Z}[C_2])$ is isomorphic to the ideal ...
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Factorization of element in quadratic integer ring

Let $K=\mathbb{Q}(\sqrt{d})$ for square-free integer $d$ and $\mathcal{O}_K$ be the integral closure of $\mathbb{Z}$ in $K$. Assume that $(a)=P_1\ldots P_n$ for prime ideals $P_i$ such that $N(P_i)\...
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Class number upper bound for imaginary quadratic fields

For real quadratic fields, there is the bound $$h\leq \lfloor\sqrt{\Delta}/2\rfloor$$ Is there anything similar for imaginary quadratic fields? More generally, I'm interested in a bound for $h$ ...
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257 views

1-1 correspondence of class group of an order '$\mathcal{O}$' and elliptic curves having complex multiplication by $\mathcal{O}$

I came across these two results Let $\mathcal{O}$ be an order in an imaginary quadratic field.There is a 1−1 correspondence between the ideal class group $C(\mathcal{O})$ and the homo-thety classes ...
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Relating the class number of a field, and of its normal closure

Suppose I take a number field $ K $, not necessarily Galois, with class number $ h_k $ (over $ \mathbb{Q} $). Write $ \overline{K} $ for the normal closure of $ K $. What, if anything, can be said ...
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Representative of the $2$-Sylow subgroup of an ideal class group

Let $C$ be the ideal class group of $\mathbb{Q}(\sqrt{-6})$. I already showed that the ideal $(2,\sqrt{-6})\in C$ is not principal in $\mathbb{Q}(\sqrt{-6})$, but it is principal in $\mathbb{Q}(\sqrt{...
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Computing ideal class group by other means than the Minkowski bound?

When calculating the ideal class group of a number field, it is common to start with the Minkowski bound, followed by decomposing finitely many prime ideals of norm less than that bound, and finding ...
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306 views

Computing the Picard group of Number Fields in analogy with genus $0$ curves

It is possible to compute the Picard group of (some? all?) genus $0$ curves in the following manner: For concreteness let, $A = k[x,y]/(x^2+y^2-1)$ and $X = \operatorname{Spec} A$. Let $Y$ denote $\...
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Ramification of primes under cetain conditions

Let $K\subset L=K(\gamma)$, ($\gamma$ an algebraic integer) be number fields such that there exists a $k\in \mathcal O_K$ and some $n\in\Bbb N$, $\gamma^n=k$. Also, there exists an ideal $I\subset \...