Questions tagged [ideal-class-group]

The tag has no usage guidance.

53 questions
Filter by
Sorted by
Tagged with
33 views

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.
32 views

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 ...
51 views

29 views

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 ...
257 views

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 ...
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 ...
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: ...
33 views

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 ...
8 views

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-...
23 views

63 views

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 ...
76 views

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....
12 views

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 $I^n=(a)$...
28 views

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 $C(\mathcal{O}_K)$ ...
65 views

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 ...
72 views

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 ...
84 views

187 views

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 ...
37 views

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 $[\mathfrak{p}]$ ...
129 views

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 (...
67 views

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 ...
134 views