# 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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### 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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### 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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### 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 ...