Questions tagged [dedekind-domain]
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.
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Tor-dimension of $A/(a)\otimes_k B$, where $A$ and $B$ are Dedekind domains
Let $A$ and $B$ be two Dedekind domains which contain a field $k$ which is algebraically closed in both $A$ and $B$. Let $a$ be a non zero element in $A$.
What is the Tor-dimension of $A\otimes_k B$? ...
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Prime of order is regular iff its decomposition in the normalization is trivial.
It's from a statement in Algebraic Number Theory by Neukirch, page 92. Example 5 "One can show..."
Let ${o}$ be a one-dimensional noetherian integral domain and $\tilde{o}$ be its ...
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$M$ $\mathscr{R}$-submodule complete lattice $\iff$ $aL \subset M \subset a^{-1}L$, $a \in \mathscr{R}$
My question may be nestled in the sense that there might be confusion about several notions, please let me know if it is the case.
In "The Arithmetic of Hyperbolic 3-manifolds" by Maclachlan ...
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Show that $(p, \alpha)$ is a non-invertible ideal in the quadratic order $\mathcal{O}=\mathbb{Z}[\alpha]$
Let $K=\mathbb{Q}(\alpha)$ be an imaginary quadratic field. Suppose $\mathcal{O}=\mathbb{Z}[\alpha]$ is a non-maximal order such that a prime $p$ divides its conductor. We might take $\alpha=\frac{d+\...
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Equivalence of notions of localization for Dedekind domains
Let $\mathcal{O}$ be a Dedekind domain with ring of fractions $K$. Given a cofinite set of primes $X\subseteq \operatorname{Spec}(\mathcal{O})$, we can form the localization with respect to the ...
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Why is there a fractional ideal $\mathfrak b$ such that $\mathfrak a\mathfrak b=\mathfrak o$?
For personal interest, I've been working through the exercises in Algebra(by Serge Lang) to create my own solution. Then I came across an exercise on Dedekind rings.
The exercise is as follows:
Prove ...
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Question about Dedekind Domains and Valuations
I've been trying to prove this claim that I think is true, but I'm getting more and more sure it's not true. Here's the set up: Let $\mathcal{O}$ be a Dedekind domain, and $K$ its field of fractions. ...
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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 ...
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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 ...
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A definition for "non-congruency" of two ideals module another, in a ring of integers of a number field.
Let $F$ be a number field and let $O_F$ be its ring of integer. I am looking for a definition for the "non-congruency" of two ideals of $O_F$ module another, in an attempt to generalize ...
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Noetherian $1$-dimensional domain with ideal generated by more than $2$ elements
It is a classical theorem that in a Dedekind domain, every ideal can be generated by two or fewer elements. I was wondering what goes wrong if we omit the integrally closed property. I thought about, ...
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Clarification on Exponents in Prime Factorization of Ideals in Dedekind Domains and Number Fields
Let $R$ be a Dedekind domain and $I$ a proper ideal. Then I know $I$ can be expressed uniquely as a finite product of prime ideals:
$$
I = \prod_{\mathfrak{p} \text{ prime}} \mathfrak{p}^{n_{\mathfrak{...
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Doubts about this lemma (Fractional Ideals and Projective Modules)
This is from Conrad's notes.
I agree that we can promote $\frak{a}$ and $\frak{b}$ from fractional ideals to integral ideals by multiplying by some element in $A$. i.e. For each prime ideal $\frak{p}$...
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Neukirch's Proof of $\dim_k (\mathfrak{O}/\mathfrak{pO}) = [L:K]$
The setup is as follows. Let $\mathfrak{o}$ be a Dedekind domain, $K$ its field of fractions, and $L/K$ a finite separable field extension with $n = [L:K]$. Furthermore, let $\mathfrak{O}$ denote the ...
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Cox's exercise 5.1 on (eventually) proving ring of integers is a dedekind domain
I am reading Cox's "Primes of the Form $x^2 + ny^2$", and am on Chapter 5 (it's a speedrun from number fields to Hilbert's class field). I am attempting exercise 5.1, and I have done some ...
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Do we have $J(I : J) = I$ in a noetherian domain?
Let $R$ be a noetherian domain and $K$ its fraction field.
Let $I$ and $J$ be fractionals ideals.
Do we have
$$ J(I : J) = I $$
where ${(I : J) = \{x \in K \vert xJ \subset I\}}$?
We clearly have $I(I ...
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Coordinate Rings and Dedekind Domains
I am relatively new to ring theory, though I have an idea of what things kind of are. I have been learning about Dedekind domains, which are integral domains which are Noetherian, integrally closed ...
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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 ...
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Why is the class group torsion?
While wandering through math stack exchange I found an interesting question, namely this one:
Why are the algebraic integers a Bezout Domain?
(Found here: Is there an elementary way to prove that the ...
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Find all ideals contains power of prime ideals
Let $R$ be a Dedekind domain.
Let $P$ be a prime ideal of $R$. Can we figure out all ideals which satisfies $P^n⊂I⊂R$ ?
Factorlization of $I$ into maximal ideals should help, but I don't exactly what ...
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$I^{-1}M \cong Hom_R(I,M)$, $R$:Dedekind domain, $I$:fractional ideal of $R$, $M$:torsion free $R$module [duplicate]
Let $R$ be a Dedekind domain, let $I$ be fractional ideal of $R$.
Let $M$ be torsion free $R$module.
Then I want to prove $I^{-1}M \cong Hom_R(I,M)$.
Let $ \phi : I^{-1}M→Hom_R(I,M)$ be given by $x→\...
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Degree of a separable extension of a Dedekind ring's quotient field
I'm studying "Algebraic number theory" written by S. Lang. In proposition 21 of chapter 1.7 it says if $A$ is a Dedekind ring and $K$ its quotient field, for a separable extension $L/K$, $[L:...
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the sum of the products of ramification degrees and relative degrees
I am reading Algebraic Number Fields by Gerald Janusz and I get confused about the part in the picture below.
Consider two Dedekind domains $R\subset R'$ with quotient fields $K\subset L$. Let $p$ be ...
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Is the ring of formal power series $\mathbb K[[x_1, \dots, X_n]]$ in $n$ indeterminates over a field a PID?
Is the local ring of formal power series $\mathbb K[[x_1, \dots, X_n]]$ in $n$ indeterminates $x_1, \dots, x_n$, $n>1$, over a field $\mathbb K$ a principal ideal domain (PID)? This is true when $n=...
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Splitting field of $x^4+2$
While learning Galois theory, I tried to construct a splitting field for the polynomial $x^4+2$ over $\mathbb{Q}$, but I am terribly stuck.
Since $x^4+2$ is irreducible by Eisenstein's criterion, I ...
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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 ...
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Solving the equation $xy=z^n$ in a Dedekind domain
Let $xy=z^n$ where $x$, $y$ and $z$ belong to a Dedekind domain $R$, with $n>1$, and $(x,y)=1$. We can also assume that the ideal class group of $R$ is torsion-free.
Then I’d like to show that $x=...
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Ramification Groups - explicitly embedding the first factor group (Marcus Number Field; Chapter 4 Exercise 21)
I've been working through some of the problems of Marcus' Number Fields and am stuck on a problem relating to ramification groups.
Let $K$ be a number field, $L$ is a normal extension of $K$, $G$ is ...
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$(\alpha, \beta)^n=(\alpha^n, \beta^n)$ in Dedekind domains
It is a sometimes useful lemma that if $\mathfrak{a}=(\alpha, \beta)$ is an ideal of a Dedekind domain $A$, then $\mathfrak{a}^n=(\alpha^n, \beta^n)$. Of course, this is easy to prove, but I'd like to ...
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Proof of local definition of Dedekind Domain without using unique factorization of ideals
We say that an integral domain $A$ is a Dedekind domain if:
$A$ is Noetherian,
$A$ is integrally closed,
$\dim A = 1$ (in other words, every prime ideal is maximal).
I would like to show that ...
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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(\...
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Defining invertible ideal as projective module of rank one contained in field of fraction
For a Dedekind domain $A$ such that $\operatorname{frac}(A)=K$, we define invertible ideals to be $I\hookrightarrow K$ and rank one projective modules over $A$. (I think this definition is motivated ...
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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 ...
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$M_p \cong N_p$ for all prime ideals $p$, but $M \not \cong N$
I am asked to find an example of a ring $R$ and $R$-module $M$ and $N$ such that $M_p \cong N_p$ for all prime ideal $P$ in $R$ but $M$ is not isomorphic to $N$.
My idea is as follows: Recall that if $...
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Prove Set is a Dedekind Cut
Consider: Let $y \in \mathbb{R}. Q_y = \{q \in \mathbb{Q} \mid q < y\}$. I would like to prove that $Q_y$ is a Dedekind cut, following the definitions below.
$d \neq \mathbb{Q}$ and $d \neq \...
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Prove Set is Dedekind Cut
Let $G$ be a Dedekind cut. Show that
$H = \{x \in \mathbb{Q} : $There exists $a \in Q_{>0}$ such that $-x-a \not \in G \}$
Prove that $H$ is a Dedekind cut.
I'm not entirely sure where to begin ...
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general theory of ideal class groups
I'm looking for good references for ideal class groups.
I have studied ideal class groups in two different context; in Algebraic number theory and in Quaternion Algebra (as they have connections with ...
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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 $\mathbb{Z}\left[\frac{1+\sqrt{65}}{2}\right]$...
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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 \...
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Inert prime ideal in a given field extension
Let $A$ be a Dedekind domain, $K$ its field of fractions. Let $L/K$ be a finite separable extension and $B$ its ring of integers. Further, let $\theta \in B$ be an integral primitive element of $L$ ...
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Factorization of primes in an algebraic number field to which Dedekind-Kummer does not apply
I am mainly using these notes on algebraic number theory. Let $L/K$ be a finite degree field extension given by $L=K(\alpha)$ (this is not restrictive by the primitive element theorem). Let $\mathfrak{...
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Doubt regarding the two definitions of Norm of an ideal in Algebraic Number theory
Let $O_L$ and $O_K$ be Dedekind domains and Let $L$ and $K$ denote their corresponding field of fractions. Further, let $ L/K $ be a finite separable extension.
For a fractional ideal $J$ of $O_L$, we ...
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Arithmetic structure including both unique factorization and Dedekind domains
Has an algebraic arithmetic structure been defined on integral domains, which would include both Dedekind rings and unique factorization domain with respect to the arithmetic properties, and more ...
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Dedekind's Theorem
Use Dedekind’s Theorem to factorise the following principal ideals in the ring of integers of the following fields.
a) $Q(√3): ⟨2⟩,⟨3⟩,⟨5⟩,⟨30⟩$
b) $Q( ^3√2): ⟨7⟩, ⟨29⟩, ⟨31⟩$
Here is what I ...
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Resolutions over Dedekind Domains
I want to prove that if $R$ is a Dedekind Domain, then $\operatorname{Ext}_R^n(M,N)=0$, for $n \geq 2$.
Then, I have a question: There are any properties about projective or injective resolutions over ...
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Is discrete valuation field a fraction field of some Dedekind domain?
Is discrete valuation field a fraction field of some Dedekind domain ?
Let $K$ be a discrete valuation field, does there exist some Dedekind domain $R$ such that $\operatorname{Frac}(R)$＝$K$ ？
If ...
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Proposition 8.3, Neukirch- Algebraic Number Theory
Let $B/A$ be a extension of Dedekind rings, where $B$ is the integral closure of $A$ in a finite and separable algebraic extension $L$ of the fraction field $K$ of $A$. Let $\theta$ be a primitive ...
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Examples of modules that $M_\mathfrak{p} \subset N_\mathfrak{p}$ that does not imply $M \subset N$?
I came across a theorem in algebraic number theory:
Theorem Let $A$ be a Dedekind ring and $M, N$ two modules over $A$. If $M_\mathfrak{p} \subset N_\mathfrak{p}$ for all prime ideals $\mathfrak{p} \...
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sieve of Eratosthenes generalization to Dedekind domains or even PID's
I'm Interested in finding irreducibles in Dedekind domains, (and especially integer rings) in an efficient manner.
I've tried to look around a bit but found no papers on this (admittedly my paper ...
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The orders of points in the function field of elliptic curves
Let $K$ be the function field of an elliptic curve $C$ over a finite field $\mathbb F_q$, and $|C|$ be the set of closed points of $C$, i.e. the set of places of $K$.
Let $A$ be any element of ...
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