All Questions
Tagged with ideals dedekind-domain
61
questions
4
votes
1
answer
125
views
Counterexample of Divisibility of Ideals with Product of Ideals
Given a commutative ring $R$ with unity, we define for $I,J\subseteq R$ ideals
$I\ \vert\ J\iff I\supseteq J$
$IJ=\{\sum_i a_ib_i:a_i\in I, b_i\in J\}$
For every commutative unitary ring $R$ it ...
- 310
1
vote
0
answers
51
views
Inverse of integral ideal in a Dedekind domain is a specific fractional ideal
Let $R$ be a Dedekind domain with fraction field $k$. Incan show that if $p$ is a prime ideal, then $p^{-1}=\{x\in k \mid xp\subset R \}$ satisfies $pp^{-1}=R$. Moreover, if $I$ is any integral ideal ...
- 1,334
1
vote
1
answer
61
views
Question regarding divisibility with prime ideals
While I was reading this paper Lenstra page 15, for d=4729494, he says $2162+\sqrt d$ is divisible by the cube of the prime ideal $(5,2+\sqrt d)$. Can someone please help me how this works?
I see that ...
- 33
2
votes
0
answers
66
views
Neukirch's Proof of $\dim_k (\mathfrak{O}/\mathfrak{pO}) = [L:K]$
The setup is as follows. Let $\mathfrak{o}$ be a Dedekind domain, $K$ its field of fractions, and $L/K$ a finite separable field extension with $n = [L:K]$. Furthermore, let $\mathfrak{O}$ denote the ...
- 14.2k
0
votes
0
answers
110
views
Theorem $17$, Chapter $3$ (Marcus' Number Fields): Every ideal in a Dedekind domain is generated by at most two elements
I understand that similar questions have been asked before, but I am looking for an explanation of certain steps in Marcus' proof of the same, in Theorem $17$, Chapter $3$ of Number Fields. I shall ...
- 14.2k
1
vote
1
answer
79
views
Can the decomposition of a principal ideal in a Dedekind domain contain a non-principal ideal as a factor?
If $I= \prod_{P_i\in Spec(R)}P_i$ is the decomposition into prime ideals of a principal ideal $I$ in a Dedekind domain $R$, can one of the $P_i$ be a non-principal ideal? I guess it can’t, but I don’t ...
- 79
1
vote
1
answer
139
views
If $I$ is an invertible ideal, then there is $\alpha$ with $N(\alpha I^{-1})$ coprime with $N(I)$
Let $\mathcal{O}$ be an order inside a quadratic number field (not necessarily maximal). I want to show that if $I$ is an invertible $\mathcal{O}$-ideal, then there is $\alpha \in I$ such that $N(\...
- 617
0
votes
2
answers
582
views
Divisibility of Ideals [closed]
I'm an undergraduate student, who's in my final semester in university.
I have a research project, but the advisor isn't the best.
He said why won't we develop the notion of divisibility of ideals in ...
- 27
0
votes
0
answers
43
views
Number of ideal classes of $\mathbb{Z}\left[\frac{1+\sqrt{65}}{2}\right]$ [duplicate]
Let $K=\mathbb{Q}(\sqrt{65})$, whose number ring is $\mathcal{O}=\mathbb{Z}\left[\frac{1+\sqrt{65}}{2}\right]$.
I know that the number of ideal classes in $\mathbb{Z}\left[\frac{1+\sqrt{65}}{2}\right]$...
- 2,067
2
votes
1
answer
136
views
Show that $aI=bJ$ where $I,J$ are ideals in $\mathcal{O}_K=\mathbb{Z}[\sqrt{-6}]$
Let $K=\mathbb{Q}(\sqrt{-6})$, and therefore $\mathcal{O}_K=\mathbb{Z}[\sqrt{-6}]$. I have already proved that $[I]=[J]$, where $I=(2,\sqrt{-6})$ and $J=(3,\sqrt{-6})$. Now I want to find $a,b \in \...
- 2,067
1
vote
0
answers
108
views
Let $A, B, C$ be ideals in a Dedekind domain $R$. Showing $A \cap (B+C) = (A \cap B) + (A \cap C)$
Let $A, B, C$ be Ideals in a Dedekind domain $R$. I want to show that $A \cap (B+C) = (A \cap B) + (A \cap C)$, where $(A \cap B) + (A \cap C) \subset A \cap (B+C)$ is obviously true. Now I would like ...
- 13
1
vote
2
answers
278
views
Ring of Formal Laurent Series which are Dedekind domains
Let $R$ be an integral domain and $R((x))$ be the ring of formal Laurent series over $R$. (The answer to this question has a good explanation for our ring.)
Is it true that $R$ is a Dedekind domain ...
- 207
1
vote
0
answers
26
views
Ramification indices and residue class degrees of $\mathfrak{O}_K$ where $K=Q[\alpha]$, $f(\alpha)=0$, and $f(x)=x^3-x-1$.
I already know that $\alpha^3-\alpha-1=0$ implies that $\{1,\alpha, \alpha^2 \}$ creates an integral basis for $\mathbb{A} \cap \mathbb{Q}[\alpha]$. I'd like to try to use Dedekind's theorem in some ...
- 123
3
votes
1
answer
691
views
Doubt in finding an inverse of an ideal
Background
I'm currently working on problem 8.7 from Alaca & Williams' Intro. Alg. Number Theory. The problem states the following:
Show that $\langle3,1+2\sqrt{-5}\rangle\mid\langle1+2\sqrt{-5}\...
- 1,162
2
votes
0
answers
458
views
GCD of Ideal: How we get $\gcd(I, J) = I + J $?
Take any two non-zero ideals $I$ and $J$ in $R$. Since we know that ideals in a Dedekind domain factors uniquely into prime ideals
$$I = \prod_i P_i^{m_i}, J = \prod_i P_i^{n_i}$$
where $P_i$’s are ...
