# Tag Info

## Hot answers tagged dedekind-domain

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### Prime ideal factorization in non-monogenic ring of integers ($K=\mathbb{Q}(\sqrt[4]{24})$)

The Kummer-Dedekind theorem does not require $\mathcal O_K = \mathbf Z[\alpha]$. If $K = \mathbf Q(\alpha)$ where $\alpha$ is an algebraic integer with a minimal polynomial $f(x) \in \mathbf Z[x]$, ...
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### Why do we want Dedekind rings to be integral closed?

One of the things that you want in Dedeking rings is that you have nice ideal factorisation properties. In particular, multiplication by a nonzero ideal should be injective. Suppose $R$ is a ...
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### Integral ideals intersected with $\mathbb Z$

I'm not sure the other inclusion holds. For example let $K=\mathbb{Q}\left(\sqrt[3]{2}\right)$. Then we have that $\mathcal O_K = \mathbb Z\left[\sqrt[3]{2}\right]$. Now consider how $5\mathcal O_K$ ...
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### Exact Sequence with Ideal Class Group

You give the example of an exact sequence defining the class group of a Dedekind domain and you wonder "So what? What do we learn now from it?" At this stage, your question is quite legitimate. Apart ...
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### An example of prime ideal $P$ in an integral domain such that $\bigcap_{n=1}^{\infty}P^n$ is not prime

Let $k$ be any field. Set $$R = \bigcup_{n=1}^{\infty} k\left[x,\ y,\ x^{1/n} y^{1/n} \right].$$ Each one of the terms in the union is a domain, so the rising union is also a domain. Let $P$ be the ...
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### Are integrally closed domains Dedekind domains?

No. Among many characterisations, Dedekind domains are noetherian integrally closed domains of Krull dimension $1$, i.e. every non-zero prime ideal is maximal. As a simple counter example, if $K$ is ...
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### Every Ideal $\mathfrak{a}$ of a Dedekind domain $\mathcal{O}$ can be generated by $2$ elements

If $\mathfrak{a}=(0)$ or $(1)$ then $\mathfrak{a}$ is principal, so assume that $\mathfrak{a}\neq (0)$ or $(1)$. Then there is some $a\in\mathfrak{a}$ which is non-zero and not a unit. By what you ...
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### Example of a UFD that is not Dedekind

If UFD $D$ which is not a field (for example $\mathbb{Z}$), then $D[X]$ is a UFD which is not Dedekind.
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### Showing that $x^2+5=y^3$ has no integer solutions.

$\mathfrak p\mid(2\sqrt{-5})$, so $\mathfrak p\mid(2)$ or $\mathfrak p\mid(\sqrt{-5})$. Since $(2),(\sqrt{-5})$ are prime, $\mathfrak p$ is equal to one of them. We can't have $\mathfrak p=(2)$, since ...
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### Is it possible for a finite integral closure of a DVR to not be a PID?

I think the following more general statement is true. Proposition. Let $A$ be a one-dimensional noetherian semi-local domain and let $K$ be its field of fractions. Consider a finite algebraic field ...
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### Factorising the ideal $(14)$

How exactly do you know that none of $\langle 2 \pm \sqrt{-10} \rangle$, $\langle 2 \rangle$, $\langle 7 \rangle$ are prime? They're not prime because those are principal ideals that are generated by ...
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If $char(A)=0$ : $N(I) \le d$ implies $A/I$ is a quotient of $A/(d!)$, your assumption is that it is a finite ring with finitely many quotients thus only finitely many ideals with $N(I) \le d$ If \$...