# 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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### Find Convolution polynomial ring $D[X]/\langle X^{N-1}\rangle$ where $D$ is Dedekind Domain] and $N$ is prime [closed]

Kindly tell me how to generate elements of this convolution polynomial ring
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### I'm looking for Dedekind domaina charichtaristics proofs [closed]

so I have been studying a bit of commutative algebra, and found 4 or 5 equivalent statements to the definition of a dedekind domain, but the proof wasn't that clear to me. Do you suggest any good ...
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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 ...
1 vote
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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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### 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 ...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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 ... 149 views

### 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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### Is $\mathbb{C}[X, Y] / (X^5 + Y - 13)$ integrally closed?

I want to check if $R := \mathbb{C}[X, Y] / (X^5 + Y - 13)$ is a Dedekind domain or not. I know $R$ is an integral domain because $X^5 + Y - 13$ is prime in $\mathbb{C}[X, Y]$. $R$ is also noetherian ...
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### Proof that $\frac{\mathbb{C}[x,y]}{\langle x^2+y^2-1\rangle}$ is a Dedekind Domain

I want to prove that $S=\frac{\mathbb{C}[x,y]}{\langle x^2+y^2-1\rangle}$ Is a Dedekind Domain. Hence I want to prove that $1$. $S$ is Noetherian, $2$. $S$ is integrally closed, and $3$. in $S$ every ...
1 vote
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### Theorem 9.8 of Atiyah : why does it suffice to show for integral ideal in Ap?

I'm trying to prove => part of atiyah 9.8. I understood everything but Can somebody help me to understand why does it suffice to show for integral ideal of Ap?
1 vote
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### 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 ...
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### Is the ring of integers of a DVR also a DVR? [closed]

$R$ is a DVR and $L$ is a finite separable extension of $Q(R)$. Can we say that the ring of integers is a DVR? I know it's a Dedekind domain. So showing it's local is enough. If it's not a DVR, what ...