1
vote
0answers
51 views

simple question regarding isomorphisms relating to ring of integers

I have a simple question about isomorphisms and ideals. Let $\mathcal O_F$ be the ring of integers in some quadratic number field $F=\mathbb{Q}(\sqrt d)$ and let $f(x)$ be the minimal polynomial of ...
15
votes
0answers
237 views

Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
0
votes
2answers
60 views

The action of a Galois group on a prime ideal of a Dedekind domain

This is a slight variant of a question I asked earlier. Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let ...
3
votes
1answer
93 views

Classification of nonzero prime ideals of $\mathbb{Z}[i]$

I know the classification of Gaussian primes: let $u$ be a unit of $\mathbb{Z}[i]$. Then the following are all Gaussian primes: 1) $u(1+i)$ 2) $u(a+ib)$ where $a^2+b^2=p$ for some prime number p ...
3
votes
1answer
166 views

Norm of ideals in quadratic number fields

I do not really understand how to factor ideals in a quadratic field $K = \mathbb{Q}(\sqrt{d})$, mainly because I have some trouble computing the norm of ideals. I think I understand what is going on ...
4
votes
1answer
149 views

Solving $x^2+19=y^5$

I was given several exercises and there is a particular one, I am not able to solve. Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
3
votes
1answer
154 views

Prime ideal decomposition in quadratic field extensions

Once you have the character $\chi$ of a quadratic field extension and the corresponding modulus $N$, it is easy to see which prime ideals split, ramify and are inert by looking at their remainder ...
2
votes
0answers
38 views

Finding reduced quadratic numbers and principal ideals

Hello :) I want to compute alle reduced quadratic numbers with discriminat $65$. We call a number $\gamma$ reduced if $\gamma>0$ and $-1<\gamma'<0$. We are working in quadratic field ...
2
votes
0answers
155 views

Find all ideals with given norm

I'm working through a final exam from 2 years ago. First task was to find the ideal class group of $\Bbb{Q}(\sqrt{-73})$. That is not the difficult work. I can give the 4 representants of the group by ...
3
votes
1answer
280 views

Diophantine equation (use class ideal group to solve)

Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$ My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
5
votes
1answer
157 views

Quotient of ring of integers

Let $R=\mathcal{O}(K)$ be the ring of the integers of $K=\mathbb{Q}[\zeta_8]$, where $\zeta_8=e^{2\pi i/8}=\sqrt{2}/2(1+i)$ is a primitive eighth root of unity in $\mathbb{C}$. It can be shown that ...
11
votes
5answers
474 views

How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
6
votes
1answer
396 views

How to show this ideal is not principal

I have been brushing up on cubic number fields. Specifically, let $s$ be a root of the polynomial $x^3 + x^2 + 3x + 17$, and consider $K = \mathbb{Q}(s)$; we have $\mathcal{O}_K = \mathbb{Z}[s]$, and ...
1
vote
0answers
198 views

Easiest way to prove that a subset of even integers is closed under multiplication?

What's the easiest way of showing that; $2\mathbb{Z}\setminus (4n-2)\mathbb{Z}$ is closed under multiplication? (I'm trying to show that $(4n-2)$ is a prime element of $2\mathbb{Z}$ by showing ...
-1
votes
1answer
414 views

Doubt on class group

I started reading Class group after some one's advice ,so I got the following doubts,I would be happy if someone clarify the doubts, I understood that the class group measures the failure of the ...
16
votes
8answers
1k views

Intuition behind “ideal”

To briefly put forward my question, can anyone beautifully explain me in your own view, what was the main intuition behind inventing the ideal of a ring? I want a clarified explanations in these ...
8
votes
2answers
1k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...