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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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Factorising the ideal $(14)$

I wish to find the prime factors of the ideal $(14)$ in $\mathbb{Q}(\sqrt{-10})$. My working so far has been by noticing that $$14=(2+\sqrt{-10})(2-\sqrt{-10})=2\times7$$ So we have the candidates $...
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Is it possible for a finite integral closure of a DVR to not be a PID?

Suppose that we have a point $A$(local ring, DVR) of an abstract curve over $k=\bar{k}$ given by a field $k(X)$. Let $k(Y)$ be a finite extension of $k(X)$ and denote by $B$ the integral closure of $A$...
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Is $\mathbb{Z}[\sqrt{-5},1/10]$ a PID

Let $R = \mathbb{Z}[\sqrt{-5},1/2,1/5]$. Inverting the ramified primes $2,5$ simplifies the proof that every maximal ideal is inversible ie. the unique factorization in maximal ideals. In $O_K=\...
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unique factorisation of non-zero fractional ideals in a Dedekind domain

I'm reading about Dedekind domains from Serre's book, and on Pg. 12, before stating Proposition 7, there are arguments for the proof, which read as follows: If one considers the ideal $a_1 = \...
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Ideals of Dedekind rings are projective

Let $R$ be a Dedekind domain and $I$ be an ideal of $R$. Show that $I$ is a projective $R$-module. My definition of a projective module is that it is a direct summand of a free module, i.e. there ...
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Is every local flat extension of a DVR a domain?

The title pretty much says it all. I would like to know if it is true that given a finite flat ring extension $A \rightarrow B$ with $A$ a DVR, and $B$ a local ring, then $B$ is necessarily a domain....
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Torsion Free Module over Dedeking Ring

Let $\phi: R \to A$ be a finite morphism of Dedekind rings (so $A$ is a finitely generated $R$-module) and $M$ a finitely generated $A$-module. Obviously, if we restrict the action of $A$ on $M$ to $R$...
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A proof for Kummer's lemma using only Dedekind domains

I need to prove this lemma : Kummer's lemma :1 But all the proofs i've seen in books require field theory ! I've tried to use this theorem to prove it , using the fact $\mathbb{Z}[\epsilon]$ is a ...
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Can every maximal ideal of Dedekind domain be principal after restricting to a small enough distinguished open subset?

Let $S=\textrm{Spec }R$ where $R$ is a Dedekind domain, let $\mathfrak{p}$ be a maximal ideal of $R$, which is a closed point of $S$, can we find an distinguished open affine subset of $S$, say $\...
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In an integrally closed, Noetherian, local, integral domain of dimension $1$, the maximal ideal $P$ is eventually principal

Let $R$ be an integrally closed, Noetherian, local, integral domain of dimension 1 with unique maximal ideal $P$. Take an element $a \in P$ that is non zero. Show that for some $n$, $P^n$ is ...
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Does any fractional ideal of $R$ always contain a non-zero element of $R$? [closed]

Let $R$ be an integral domain. Let $A$ be a non-zero fractional ideal of $R.$ Then can we say that $A$ always contains a non-zero element of $R$? Please help me in this regard. Thank you very much.
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Understanding old question proof about Dedekind domain

Let $R$ be a Dedekind domain. Given two non-zero ideals $\mathfrak a, \mathfrak b$ in $R$ there exists $c\in K^\times$ (the fraction field of $R$) such that $c\mathfrak a$ and $\mathfrak b$ are ...
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Powers of maximal ideal in Dedekind ring

Suppose $K$ is a number field (you may suppose $K$ is imaginary quadratic if necessary, but I doubt that matters) with ring of integers $A$, and suppose $\mathfrak{p}$ is a prime of $A$. Choose $t, u\...
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Splitting primes and principal ideals

Let $\mathbb{Q} \subseteq K\subseteq L$ be finite extensions of $\mathbb{Q}$. Suppose that $P$ is a prime ideal of $\mathcal{O}_K$, and that $P$ is not principal. Does this automatically imply that ...
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Flat scheme over a Dedekind ring

I have a problem with Proposition 4.3.9 of Qing Liu's algebraic geometry book. It says if R is a Dedekind ring and X is a reduced scheme and we have a dominant morphism $f:X\to \operatorname{spec}R$, ...
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Division of ideals in a Dedekind Domain [duplicate]

If $K \subset L$ are number fields and $I$ and $J$ are ideals of $\mathcal{O}_K$, then I'm trying to prove the statement $$I\mathcal{O}_L | J\mathcal{O}_L \implies I|J,$$ where the first division ...
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Application Nakayama's Lemma for Field Extensions $L \vert K$

I have a question about an argument used in Tamas Szamuely's "Galois Groups and Fundamental Groups" in following excerpt (see page 96): According the proof we firstly show that $\sum^r _{i=1} e_i [\...
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Prove that the ring $\mathcal O_K$ of algebraic integers of $K = \Bbb Q (\sqrt d)$ is a Dedekind domain.

Prove that the ring $\mathcal O_K$ of algebraic integers of $K = \Bbb Q (\sqrt d)$ ($d$ is a square free integer) is a Dedekind domain. I have taken an ideal $I \subseteq \mathcal O_K$. Consider the ...
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Showing that $x^2+5=y^3$ has no integer solutions.

I'm trying to show that the Diophantine equation $x^2+5=y^3$ has no integer solutions using the fact that $\mathbb Z[ \sqrt{-5}]$ has class number two. I think I have the general idea, but I'm having ...
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Torsion module Finite composition length

Let R be a Dedekind domain. A be a finitely generated R-module. Then A = A 1 ⊕ A 2 , for some torsion module A 1 and torsion-free module A 2. Proof that A1 has finite composition length. I can see ...
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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 $I^n=(a)$...
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about CRT theorem [duplicate]

Let $R$ be a Dedekind ring and assume that the prime ideals are $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$. Then $\mathfrak{p}_1^2,\mathfrak{p}_2,\ldots,\mathfrak{p}_n$ are coprime. Pick an element $\pi \...
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about Dedekind domain

Let $A$ be a Dedekind domain, $S$ a multiplicatively closed subset of $A$. Show that $S^{-1}A$ is either a Dedekind domain or the field of fractions of $A$. Attempt: $A$ is an integrally closed ...
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Ideal lying over an ideal shows up in prime decomposition

Say we have a number ring $R$ of a number field $K.$ We know that $R$ has unique prime ideal decomposition. My question is as follows. Let $I \subset J$ be a containment of ideals. Then let $I = P_1 \...
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A Question from Neukirch Regarding the Quotient Module of a Sum of Modules

The Setup: Let $L/K$ be a separable extension with rings of integers $\mathcal{O}_{L}$ and $\mathcal{O}_{K}$ respectively. Let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_{K}$ and $\mathfrak{p}\...
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determine the set of prime ideals of the Dedekind Domain

Determine the set of prime ideals of the Dedekind Domain $\mathbb{Z}[i] = \{a+bi: a,b \in \mathbb{Z}\}$ and factor the ideal $(7-i)$ into prime ideals. I know in Dedekind Domains only proper ideals ...
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Integral ideals intersected with $\mathbb Z$

Let $I \subseteq \mathcal O_K$ be an integral ideal of a number field $K$ such that $p \nmid I$ for all primes $p \in \mathbb N$. I want to show that $I \cap \mathbb Z = N(I) \mathbb Z$. It's clear ...
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Rings over which torsion free module is projective

If R is an integral domain with the property that any finitely generated torsion free R-module becomes projective. Certainly R could be Dedekind domain. But is it necessary that R must be Dedekind ...
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On an auxiliary lemma for the structure theorem on Dedekind domains

I'm searching for a proof of the following statement Let $M$ be a $P$-torsion module over a Dedekind domain $R$, and let $x_1 \in M$ such that $\operatorname{Ann}_R(x_1) = Ann_R(M)$. If there exist ...
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If $R$ is a Dedekind domain and $I, J \triangleleft R$, then $R^d \oplus I \simeq R^{d'} \oplus J$ implies $d = d'$ and $I \simeq J$

I'm trying to figure out the following observation I have written on my notes on Dedekind domains, If $R$ is a Dedekind domain, $I, J$ two integral ideals and $d, d' \in \mathbb{N}$, such that $R^d ...
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Properties of the first syzygy over Dedekind domains

Let $L$ be a Dedekind domain and $I = (a) + (b)$ a non-principal proper ideal in $L$. Consider equation of the form $xa + yb = 0$. Is it true that then there exists a proper ideal $J$ such that any $x,...
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Why do we want Dedekind rings to be integral closed?

As I understand, the idea of Dedekind domains is motivated by the wish to factorize ideals into prime ideals. Dedekind rings are supposed to: be noetherian, which makes sense because that ensures ...
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Prime ideal in Dedekind domain

Let $\mathcal{o}$ be a Dedekind domain, $K$ its field of fractions, $L$ a finite separable field extension of $K$, and $\mathcal{O}$ the integral closure of $\mathcal{o}$ in $L$. When $\mathfrak{p}$ ...
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Is a number ring a Dedekind domain?

I'm studying the number field sieve factorization method, but I am having some trouble with the definition of number ring. I have found two different definitions: A number ring is a subring of a ...
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Fractional ideals are isomorphic to relatively prime integral ideals in Dedekind domains

Let $R$ be a Dedekind Domain with fraction field $K$ with nonzero fractional ideals $A$ and $B$ (i.e., $A=d^{-1}I$ for some ideal $I$ of $R$ and $d\in R$). $\mathbf{Problem}$. Show that there are ...
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Ideals in quotients of Dedekind domains

Let $R$ be a Dedekind Domain and $I=P^{a_{1}}_{1}\cdot\cdot\cdot P^{a_{n}}_{n}$ an ideal of $R$. I'm trying to understand the proof that every nonzero ideal in $R/I$ is principal. In particular, why ...
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Fractional ideals problem

If $R$ is an integral domain with field of fractions $K$ and $A$ is a fractional ideal of $R$ in $K$ (i.e., an $R$-submodule of $K$ such that $dA\subset R$ for some nonzero $d\in R$), then define $A'=\...
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Serre's (Krull's) criterion for normality in practice

Recall Serre's criterion for normality. Let $f=f(t),g=g(t) \in \mathbb{C}[t]$, and assume that $m:=\deg(f) \geq 2$, $n:=\deg(g) \geq 2$. Denote $f=a_m t^m+\cdots +a_1t+a_0$ and $g=b_n t^n+\cdots +...
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Characteristic of residue field in a Dedekind domain.

Let's consider a Dedekind domain $A$ with field of fractions $K$. Let $L$ be a finite Galois extension of $K$ and $B$ the integral closure of $A$ in $L$. Let $\mathfrak{p}\subset A$ be a non-zero ...
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There exist an integral ideal prime to a given nonzero integral ideal

Let $\mathfrak{m}$ be a nonzero integral ideal of the dedekind domain $\mathfrak{O}$. Show that in every ideal class of $Cl_K$, there exist an integral ideal prime to $\mathfrak{m}$. My effort : ...
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Class Ideal Group of $\mathbb{Q}(\sqrt{65})$

Let $O_K = \mathbb{Q}\left(\frac{1+\sqrt{65}}{2}\right)=\mathbb{Q}(\alpha)$ be the ring of algebraic integers of $K = \mathbb{Q}(\sqrt{65})$. I want to find the Class Ideal Group $G$ of $O_K$. The ...
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Every cut in R+ ist a dedekind cut?

I have hard time proving obvious theories. For my Analysis class they asked me to proof that ever cut in R+ is a dedekind cut. I know these are the defintions: A Dedekind cut is a pair (A, B), where A ...
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Prime decomposition in a Dedekind domain

Let $R=\mathbb{F}_5[X,Y]/(Y^2-X^3-2X)$. I have to determine the structure of the $R$-module $M=R/XR\times R/YR$ using the structure theorem for finitely generated modules over Dedekind domain. Since ...
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Is $M_n(R)$ Dedekind if $R$ is Dedekind?

Definition: A ring is Dedekind if every nonzero proper ideal factors into a product of prime ideals. Let $R$ be a ring. $M_n(R)$ denotes the ring of matrices with elements from $R$. Is $M_n(R)...
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Show that a ring is a Dedekind domain

Let $R$ be a Dedekind domain and $f \in R[X]$ irreducible over the field of fractions of $R$ such that $f$ and $f'$ generate the unit ideal. I want to show that $S=R[X]/(f)$ is also a Dedekind domain,...
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Proof that ideals of Dedekind domains are generated by at most two elements: how does the bolded step in the proof work?

Sorry for the slightly obscure title, but I could not think of a better way to word it. The lecture notes for my commutative algebra course have the following proof that Dedekind domains' ideals are ...
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How does passing to ideals solve the problem of unique factorization?

$A:=\mathbb{Z}[\sqrt{-5}]$ is not a UFD, because for instance $$21 = 3 \cdot 7 = \left( 1+2\sqrt{-5}\right) \cdot \left(1-2\sqrt{-5}\right).$$ But since $A$ is a Dedekind domain, we should have ...
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Application Chinese Remainder Theorem to Dedekind Domains

I have a question about the application of CRT in a proof of following thread: Dedekind domain with a finite number of prime ideals is principal The claim is that a Dedekind domain with a finite ...
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How a finite extension of transcendence degree $1$ extensions induces a morphism of curves

Let $k$ be an algebraically closed field. Throughout, by curve I mean integral, nonsingular, dimension $1$ scheme proper over $k$. In particular I'm assuming we're dealing with complete curves in the ...
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Localization of Modules over Dedekind domains

I have been reading the book Algebraic Number Theory by J.W.S. Cassels & A. Frohlich and I am currently stuck in a proof about Localization of Modules over Dedekind domains. Let $R$ be a Dedekind ...