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 ...
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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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Discriminant of non-degenerated Billinearform is square in Ideal Class Group

Let $A$ be a Dedekind domain, $K= Frac(A)$ its field of fractions and $V$ a $n$-dimensional vector space over $K$. A lattice of $V$ (with respect to ring $A$) is a sub-$A$-module $X$ of $V$ that is ...
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3 votes
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quotient of the ring of integers by a prime ideal

Let $\mathcal{P}$ be a prime ideal of the ring of integers $\mathcal{O}_{K}$ of a field $K$. Since $\mathcal{O}_{K}$ is a Dedekind domain therefore we can say that $\mathcal{O}_{K}/ \mathcal{P}$ is a ...
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Exercise 5.6(e) of Cox's Primes of the Form $x^2+ny^2$

I am having a lot of trouble proving part (e) on the exercise 5.6 in David Cox's book "Primes of the Form $x^2+ny^2$". The result, proposition 5.11, is a special case of Dedekind's prime ...
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Cardinality of associate classes in quotient ring of Gaussian integers.

Let $n=up_1^{n_1}p_2^{n_2}\cdots p_k^{n_k}$, where $p_i$ is a prime and $u$ is a unit in $\mathbb{Z}[i]$ be the factorization of $n$ in $\mathbb{Z}[i]$. Then for any proper divisor $d$ of $\mathbb{Z}[...
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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 ...
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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?
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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 ...
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gcd of $6(1-\sqrt{-5})$ and $3(1+\sqrt{-5})(1-\sqrt{-5})$doesn't exist in $\Bbb Z[\sqrt{-5}]$

gcd of $a=6(1-\sqrt{-5})$ and $b=3(1+\sqrt{-5})(1-\sqrt{-5})$doesn't exist in $\Bbb Z[\sqrt{-5}]$. Here we can see that $N(a)=216$ and $N(b)=324$. On a contrary if the gcd exists and is $d$ say then $...
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Is every element in the class group is represented by a prime ideal?

Let $K$ be a real quadratic number field and $\mathcal O_K$ its ring of integers. Is it known whether for each element in the class group we have a representative $\mathfrak p \subset \mathcal O_K$ ...
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2 votes
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Is it possible to find a product representation $xy=\varepsilon-1$?

Let $K$ be a real quadratic number field and $\mathcal O_K$ its ring of integers. Let $\mathfrak a \subset K$ be a fractional ideal and $\varepsilon \in \mathcal O_K^\times$. My question: Can $\...
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Proposition about localizations of Dedekind domains - Lang: Algebraic Number Theory

This proposition is in Lang's book "Algebraic number theory": "Let $A$ be a Dedekind ring and $S$ a multiplicative subset of $A$. Then $S^{-1}A$ is a Dedekind ring. The map $\mathfrak{a}...
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Ring of Formal Laurent Series which are Dedekind domains

Let $R$ be an integral domain and $R((x))$ be the ring of formal Laurent series over $R$. (The answer to this question has a good explanation for our ring.) Is it true that $R$ is a Dedekind domain ...
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Module of differentials of an extension of Dedekind domains is cyclic

Let 𝐴 be a Dedekind domain with fraction field 𝐾, 𝐿|𝐾 a finite separable field extension and 𝐵 the integral closure of 𝐴 in 𝐿. Assume that all the residue field extensions are separable. In the ...
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Fractional ideal of a Dedekind domain A is a projective module

It can be known that any fractional ideal $\alpha$ of A can be generated by two elements, and is that enough to construct a reverse of the quotient map from $A^2$ to $\alpha$? And by the way, I wonder ...
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1 vote
2 answers
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Questions on ring fixed by automorphisms

Suppose $A$ is an integral domain and $F$ is its field of fractions. Let $G \leq Aut(F)$ be a group of automorphisms of $F$. Assume $g(a) \subseteq A$ for all $g \in G$ and let $A^G$ be the fixed ring ...
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3 votes
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Counterexample for the sum formula in Dedekind extension

Let $A\subseteq B$ be Dedekind domains, and $K\subseteq L$ be their quotient fields respectively, and assume that $B$ is the integral closure of $A$ in $L$. Let $\mathfrak{p}$ be a prime ideal of $A$....
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2 votes
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Factoring ideal into prime ideals in $\mathbb{Z}[\sqrt{-5}]$

I would like to write the ideal $(9)$ as a product of prime ideals in $\mathbb{Z}[\sqrt{-5}]$, which is a Dedekind domain. We have $$9=3 \cdot 3=(2+\sqrt{-5})\cdot (2-\sqrt{-5}) $$ and I have shown ...
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Ramification indices and residue class degrees of $\mathfrak{O}_K$ where $K=Q[\alpha]$, $f(\alpha)=0$, and $f(x)=x^3-x-1$.

I already know that $\alpha^3-\alpha-1=0$ implies that $\{1,\alpha, \alpha^2 \}$ creates an integral basis for $\mathbb{A} \cap \mathbb{Q}[\alpha]$. I'd like to try to use Dedekind's theorem in some ...
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Do Cauchy Sequences and Dedekind cuts have any real-world applications?

Do Cauchy Sequences and Dedekind cuts have any real-world applications? Is their only purpose to complete the real numbers? What are the advantages and disadvantages of Cauchy Sequences versus ...
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$[B/\mathfrak{P}^n:A/p]=\sum_{k=0}^{n-1}[\mathfrak{P}^k/\mathfrak{P}^{k+1}:A/p]=n[B/\mathfrak{P}:A/p]$

If $A$ is a Dedekind domain with field of quotients $K$, $L$ is a finite separable extension of $K$ and $B$ is the integral closure of $A$ in $L$ then it is known that $B$ is also a Dedekind domain. ...
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