Questions tagged [ideal-class-group]
113 questions
45
votes
6 answers
Motivation behind the definition of ideal class group
Let $O$ be a Dedekind domain and $K$ its field of fractions. The set of all fractional ideals of $K$ form a group, the ideal group $J_K$ of K. The fractional principal ideals $(a) = aO, a \in K^*$, form a subgroup of the
group of ideals $J_K$,…
Mohan
- 15,494
29
votes
5 answers
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 factorization if and only if it has class number…
Adithya Chakravarthy
- 2,582
12
votes
1 answer
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 $\mathcal O_K = \mathbb Z[\alpha ]$ (with $K := \mathbb…
user9620780
- 519
12
votes
1 answer
$\mathbb{Q}(\sqrt[3]{17})$ has class number $1$
Let $\alpha:=\sqrt[3]{17}$ and $K:=\mathbb{Q}(\alpha)$. We know that $$\mathcal{O}_K=\left\{\frac{a+b\alpha+c\alpha^2}{3}:a\equiv c\equiv -b\pmod{3}\right\}.$$
I have to show that $K$ has class number $1$, i.e. $\mathcal{O}_K$ is a PID. The…
user72870
- 4,252
10
votes
2 answers
noncyclic class group example of a cubic number field
I'm trying to compute ideal class groups of various number fields, and now I'm a little familiar to a class group of a quadratic number field. However, I can't find any non-cyclic example of a class group of a cubic number field. In Marcus' "Number…
Seewoo Lee
- 15,670
8
votes
1 answer
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 $h$ involving the discriminant of the number field.
Javier
- 89
7
votes
1 answer
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 the density of primes for which it is odd known…
did
- 407
7
votes
3 answers
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 using Dedekind's criterion) for each prime $p$…
probablystuck
- 726
6
votes
1 answer
Calculate the ideal class group of $K=\mathbb{Q}(\sqrt[3]{11})$
$\textbf{Calculate the ideal class group of $K=\mathbb{Q}(\sqrt[3]{11})$}$:
Let $\alpha=\sqrt[3]{11}.$ We need the fact that the ring of integer of $K$ is $\mathbb{Z}[\alpha]$.
One basis:$\{x_1,x_2,x_3\}$. Let $\theta_1=\alpha, \theta_2=\alpha w,…
Bowei Tang
- 3,763
6
votes
4 answers
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 write the element $6$ as a product of…
Adithya Chakravarthy
- 2,582
6
votes
1 answer
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 into a product of primes, does so uniquely. The…
Jason Joseph
- 71
6
votes
1 answer
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 exploiting the relations between prime ideals (p. 73). If…
Oromis
- 686
6
votes
1 answer
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 about the relation between $ h_K $ and $…
Steven Charlton
- 2,052
5
votes
2 answers
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 constructing the class group as the quotient of the…
Ben Doyle
- 83
5
votes
1 answer
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 group $\mathbb{Z_9}$. I wish to prove this for…
User
- 954