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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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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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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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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Bounding the minimal number of generators of $O_K$

For a finite extension $K/\Bbb{Q}$ of degree $n$, If $\gcd(A,B)=1,a_j,b_j\in O_K$ and $$\Bbb{Z}[a_1,\ldots,a_r]\to O_K/(A),\qquad \Bbb{Z}[b_1,\ldots,b_r]\to O_K/(B)$$ are surjective then $\Bbb{Z}[...
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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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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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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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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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Criterion for an extension of complete fields to be unramified

Let $A$ be a complete $DVR$ with fraction field $K$. Let $L/K$ be a finite separable extension of $K$ and let $B$ be the integral closure of $A$ in $L$. Then $B$ is also a complete $DVR$ and let us ...
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Dedekind domains equivalent conditions

I am trying to understand the equivalent conditions that define Dedekind Domains given in Atiyah and Mcdonald. The proof is reduced to references to previous lemmas and has gone completely over my ...
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Theorem 9.8 Atiyah and Macdonald

I don't understand why $\mathfrak b=\mathfrak a_{\mathfrak p}$. Since $\mathfrak b$ is a fractional ideal of $A$, we have that $\mathfrak b\subset Q(A)$ (where $Q(A)$ is the quotient field of $A$), ...
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Number of ideals with norm equals to a certain number in number field.

I can't follow the claim in the lecture notes below about the number of ideals with norm $n$. Any help or hint is appreciated.
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From general ramification theory to ramification indexes of morphisms of elliptic curves

The reference for what I'll say here is Silverman's The arithmetic of elliptic curves, ch. 2, § 2. Let $C_1, C_2$ be two elliptic curves, $\Phi : C_1 \to C_2$ a non trivial morphism of algebraic ...
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Localization of Dedekind domain at a prime ideal is a P.I.D

Let $A$ be a Dedekind domain and $\mathfrak{p}\subset A$ be a prime ideal. Then the localization $A_\mathfrak{p}$ is also a Dedekind domain. I can show it has a unique maximal ideal $\mathfrak{p}':=\...
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Prime ideal factorization in non-monogenic ring of integers ($K=\mathbb{Q}(\sqrt[4]{24})$)

Consider the number field $K=\mathbb{Q}(\sqrt[4]{24})$. Then one can show that $\mathcal{O}_{K}=\mathbb{Z}[\alpha,\beta]$ where $\alpha=\sqrt[4]{24}$ and $\beta=\frac{\alpha^{3}}{4}$. The ...
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Chapter 3, Theorem 24 on Marcus' “Number Fields”.

Theorem 24 of Marcus' Number Fields states the following: Let $p$ be a prime number and assume that $p$ is ramified in a number ring $\mathcal{O}_K$. Then $p\mid disc(\mathcal{O}_K)$. There is a ...
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Quotient ring Isomorphism in dedekind domain

I was reading Jurgen Neukirch and I came across this in lemma $10.1$ and $10.2$ Let $\zeta_n$ be the $n$th root of unity and $n=l^k$, $l$ being a prime, then $(\lambda)=(1-\zeta_n)$ is a principle ...
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Understanding a theorem about Dedekind domains.

I am reading "A Course in Ring Theory" by Passman. I was reading Theorem $7.8$ which state "Let $R$ be a Dedekind domain with field of fractions $F$ and let $K$ be a finite degree ...
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Doubt in finding an inverse of an ideal

Background I'm currently working on problem 8.7 from Alaca & Williams' Intro. Alg. Number Theory. The problem states the following: Show that $\langle3,1+2\sqrt{-5}\rangle\mid\langle1+2\sqrt{-5}\...
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Primes in an extension over Dedekind domain

I am considering Dedekind domain $\mathbb{Z}[a]/(a^3-a-1)$. I consider the splitting of $23\in \mathbb{Z}$. I know from Daniel Marcas, that if $Q$ is a prime dividing $23$ in the extension, them $Q\...
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How to understand whether a given ideal is a prime ideal in a Dedekind domain

$\DeclareMathOperator{\Norm}{Norm}$I have started Algebraic Number Theory, and I have a basic doubt about how to understand whether a given ideal is prime or not in a Dedekind domain. What I thought ...
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Product of primes over number rings.

Let $K \subseteq L$ be number fields, and $R= \mathbb{A}\cap K \subseteq S = \mathbb{A}\cap L$ be number rings in each $K$ and $L$. By integral extension, we knows that if $\mathfrak{q}$ in $S$ lying ...
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Ring below Dedekind domain is Dedekind domain.

Question Let $R$ be a Dedekind domain with quotient field $K$ and $L$ a subfield of $K$ such that $R$ is integral over $R\cap L$. Show that $R'=R\cap L$ is a Dedekind domain. Attempt I have shown that ...
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GCD of Ideal: How we get $\gcd(I, J) = I + J $?

Take any two non-zero ideals $I$ and $J$ in $R$. Since we know that ideals in a Dedekind domain factors uniquely into prime ideals $$I = \prod_i P_i^{m_i}, J = \prod_i P_i^{n_i}$$ where $P_i$’s are ...
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Does locally DVR implies Dedekind Domain when it is 1-dimensional, semi-local domain but Noetherian not given

Let R be a semi-local integral domain of dimension 1 such that $\forall P \in Spec{R} $ such that $P \ne 0$ we have, $R_P$ to be a Discrete Valuation Ring. Then prove that $R$ is a Dedekind Domain? ...
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Which of the following are Dedekind domains? [closed]

Could you please advise me which of the following is a Dedekind domain, which not and why? $a) \; \mathbb Z[1/3]$ $b) \; \mathbb Z[\sqrt{-5}]$ $c) \; \mathbb Z[x]$ $d) \; \mathbb C[x,y]/(y^2 - x^3 +...
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Valuation at a prime ideal

I'm having troubles understanding the following definition Can somebody help me understanding that definition? What is for example the power of an ideal?
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Is it true that if an ideal $I$ of ring $R$ can be denoted as the product of ideals $J$ and $K$ then $I \subseteq J$ and $I \subseteq K$?

I just proof-read a proof of someone, and in the proof the assumption is used that if $I$ is an ideal of a ring $R$ such that $I = JK$ for some other ideals $J$ and $K$, then $I \subseteq J$ and $I \...
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Find the composition of finitely generated module over Dedekind domain

I'm taking a course on commutative algebra and we learn this theorem: Every finitely generated module M over Dedekind domain A is direct sum of projective module P and torsion module T. T is direct ...
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Prime decomposition of pR in $\mathbb{A}\cap \mathbb{Q}[\alpha]$ for $\alpha={^3\sqrt{hk^2}}$ if p is a prime such that $p^2|m$

I'm going through Marcus number Field chapter 3 an I'm finding very hard to understand the part about the decomposition of pR (theorem 27) that tells us that if $p\not||R/Z[\alpha ]|$ then we can ...
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Structure theorem for modules over Dedekind domains

I've come across the structure Theorem for fin. gen. Modules over a Dedekind domain several times now. It was formulated to us the following way: Let $R$ be a Dedekind domain. For every element $\...
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Factorization into prime ideals in dedekind domain $\mathbb{C}[t]_{(t)}[x]/(x^3+x^2+t)$

$R = \mathbb{C}[t]_{(t)}[x]/(x^3+x^2+t)$ is a dedekind domain. Therefore every proper ideal $I$ can be written as a product of prime ideals. I want to find the factorization of $I = (t+x^2-x)$ into ...
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Finding class number of quadratic number field using Minkowski bound

My understanding of this is as follows: In the general case, one has a quadratic number field $F$, which is always of the form $\mathbb{Q}(\sqrt{d})$ for some square-free integer $d$. Minkowski ...
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Show that $R[X]/(f)$ is Dedekind domain

Let $R$ be a dedekind domain, $K$ the field of fractions of $R$ and $f \in R[X]$ irreducible as polynomial in $K[X]$ s.t. $(f,f') = (1) = R[X]$. I want to check that $S = R[X]/(f)$ is also a dedekind ...
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Localization of Dedekind Domains

I'm struggling with the following question: suppose that $K$ is a Dedekind Domain with ring of integers $\mathcal{O}_K$, and that we have an element $x\in\mathfrak{p}$ such that $x\not\in\mathfrak{p}^...

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