# Questions tagged [dedekind-domains]

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38
questions

7
votes

1
answer

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### Quotients of number fields by certain prime powers

I apologise in advance for what must be a naive question. Let $\mathcal O_K$ be the ring of integers of the algebraic number field $K.$ Let $p$ be a rational prime, and factorize $$(p)=\mathfrak p_1^{...

6
votes

1
answer

244
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### What is the right level of generality for $(R/a) \times (R/b) \cong (R/\gcd(a,b)) \times (R/\operatorname{lcm}(a,b))$?

Let $R$ be a principal ideal domain. Let $a$ and $b$ be two elements of $R$. Let $g$ be a greatest common divisor of $a$ and $b$, and let $\ell$ be a least common multiple of $a$ and $b$. (Of course, ...

13
votes

1
answer

609
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### Can a Dedekind domain have a power basis over a ring that isn't a Dedekind domain?

This aim of this question is to determine whether there exists a proof (or some counterexample) to the following statement : "If $R$ is a subring of Dedekind domain $S$, such that $S$ has a power ...

2
votes

0
answers

65
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### How to get a concrete description of $\pi_*\Omega_{X/S}(Z)|_Z$, when $X \supset Z \to S$ is a finite extension of Dedekind schemes?

Let $X/S$ be a proper, smooth relative curve over a Dedekind scheme $S$, for example, $X = \mathbb{P}^1_S \xrightarrow{\pi} S$.
Suppose that $Z \to X$ is a horizontal effective Cartier divisor such ...

1
vote

0
answers

376
views

### Examples of almost Dedekind domains that are not Dedekind

All I know about almost Dedekind domains (which I have come to learn about only recently) is that they are integral domains whose localization at every prime is a discrete valuation ring. In other ...

1
vote

1
answer

175
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### An example of a special $1$-dimensional non-Noetherian valuation domain

I am looking for a $1$-dimensional non-Noetherian valuation domain $R$ such that there exists a sequence $\{a_i\}_{i=1}^\infty$ of elements of $R$ such that $\langle a_1\rangle \subsetneqq\langle a_2\...

2
votes

0
answers

140
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### Characterization of algebraic integers providing a prime ideal

Let $\alpha$ be an algebraic integer and let $\mathcal{O}_{\mathbb{Q}(\alpha)}$ be the ring of integers of $\mathbb{Q}(\alpha)$.
Question: How to characterize an algebraic integer $\alpha$ such that $\...

1
vote

1
answer

396
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### A question about Dedekind schemes and proper morphisms

The following is Exercise 15.3 of Görtz-Wedhorn Algebraic Geometry I:
Let $S$ be a Dedekind scheme with function field $K$ and let $f: X\to S$ be a proper morphism of schemes. Then the canonical map $...

3
votes

1
answer

239
views

### Existence of algebraic integer with absolute value equal to reciprocal of maximum of $1$ and absolute value of a given algebraic number

Consider a number field $K$, and let $v_1, \cdots v_n$ ($n \in \mathbb N$) be some finite (i.e. non-archimedean) places of $K$. Is the following true?
For every $\alpha \in K^\times$ there exists $\...

0
votes

1
answer

513
views

### On $L$-function of permutation representation

I came across the statement in a book:
Let $k$ be a number field and $K$ be a Galois extension of $\mathbb Q$ containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\...

2
votes

1
answer

77
views

### Extension of Dedekind domains and their codifferent

Let $A\subset B$ be a finite extension of Dedekind domains. Let $0\neq b\in B$ and $0\neq a\in A$ such that $(a)=(b)\cap A$. In particular, we have $a=b\cdot c$ for some $c\in B$. Now for any $A$-...

6
votes

1
answer

245
views

### Is there a finite extension with a non-trivial class group of any PID?

Let $R$ be a PID with infinitely many prime ideals. Does there always exist a finite extension $R\subset R'$ with $R'$ being a Dedekind domain with a non-trivial class group?

1
vote

1
answer

147
views

### Locally isomorphic algebras over a Dedekind domain

Let $R$ be a Dedekind domain. Let $A$ and $B$ be two finitely generated domains over $R$. Assume that for every maximal ideal $\mathfrak{p}\subset R$ the $R_{\mathfrak{p}}$-algebras $A_{\mathfrak{p}}$ ...

2
votes

1
answer

135
views

### Special submodules over almost Dedekind domains

An integral domain $R$ is an almost Dedekind
domain if for each maximal ideal $m$ of $R$, the ring $R_m$ is a Dedekind
domain, where $R_m$ is the localization of $R$ at $m$.
Question: Let $M$ ...

1
vote

0
answers

186
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### Does separability of residue fields implies separability of $L/K$?

Let $A$ be discrete valuation domain, and $K$ be quotient field of $A$. Let $L$ be a finite extension of $K$ and $B$ be the integral closure of $A$ in $L$.Does separability of residue fields implies ...

1
vote

1
answer

126
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### Decomposing Noetherian hereditary rings of Krull dimension $1$ into product of hereditary domains (i.e. Dedekind domains)

Let $R$ be a commutative Noetherian hereditary ring (https://en.wikipedia.org/wiki/Hereditary_ring) of Krull dimension $1$. Then is it true that $R$ is a finite direct product of Dedekind domains ?

3
votes

1
answer

504
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### On integral domains over which special kind of modules are projective

For an integral domain $R$ let $\mathrm{Frac}(R)$ denote its field of fractions. Then $R$ is embedded in $\mathrm{Frac}(R)$ and we can consider $\mathrm{Frac}(R)$ as an $R$-module.
Can we ...

1
vote

0
answers

233
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### Quotient of polynomial ring over a Dedekind domain

Let $A$ be a Dedekind domain and let $B:=A[X]/(f(X))$, where $f(X)\in A[X]$ is some monic polynomial such that $B$ is a domain. If I take the canonical map $A\longrightarrow B$, then it nduces a ...

2
votes

1
answer

337
views

### quotient by ideals and fractional ideals

Let $A$ be a Dedekind domain, $I$ be an ideal in $A$ and let $I^{-1}$ be the inverse of $I$ as a fractional ideal in $K$, where $K$ is the quotient field of $A$. It seems quite natural to have a $A-$...

2
votes

0
answers

445
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### Decomposition and inertia groups, and reduction modulo primes

Let $R$ be a Dedekind domain with fraction field $K$, and let $\mathfrak p$ be a maximal ideal of $R$. Let $f\in R[x]$ be a monic, separable polynomial and let $N/K$ be a splitting field of $f$.
A ...

3
votes

0
answers

425
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### Quotient of Dedekind domains

Is there any characterization for a commutative ring to be a quotient of a Dedekind domain?

1
vote

1
answer

427
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### Noetherian almost Dedekind domain

A Dedekind domain is an integral domain in which every nonzero proper ideal factors into a product of prime ideals, and an integral domain $R$ is called almost Dedekind whenever $R_m$ is Dedekind ...

5
votes

1
answer

2k
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### Generalizing Dedekind's Factorization Theorem

A classical theorem due to Dedekind states the following:
Let $O_{K}$ be the ring of integers of a number field $K$, and assume
$K$ is generated by adjoining the algebraic integer $\alpha$ to
$...

1
vote

1
answer

117
views

### Freeness of the group of principal ideals of a number field

This is just a question wondering whether a concrete (enough) result that can be proved using the axiom of choice can be proved without it. The result being that the group of principal ideals $P_K$ of ...

4
votes

2
answers

397
views

### Transitivity of discriminant for flat algebras

Sorry if the question doesn't feed this site, I'm reposting it from MSE. Nobody answered it there and I couldn't find the proof in general case(whenever it was mentioned the proof was referred to as a ...

6
votes

3
answers

766
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### Algebraic characterization of commutative rings of Krull dimension 1,2, or 3

A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about the algebraic structure of zero-...

3
votes

1
answer

303
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### Idempotent fractional ideals of a Noetherian domain

Let $R$ be a commutative Noetherian domain, $K$ its fraction field, and $J$ a fractional ideal (i.e. a finitely generated sub-$R$-module of $K$) such that $J^2=J$. Is it true that $J=0$ or $J=R$? If ...

2
votes

1
answer

218
views

### Simultaneous decomposition of modules over Dedekind domains

I posted the question on mathexchange as well, but realized that my chances would be higher posting here;
In the paper "Almost diagonal matrices over Dedekind Domains" by L. Levy a specific ...

8
votes

2
answers

802
views

### Number of ways to write an integer as a product of irreducibles

Is there any way to tell the number of distinct ways to factor $a\in\mathcal{O}_k$ (up to units, of course) when $k$ is not a PID? A simple investigation in $\mathbb{Q}(\sqrt{-5})$ with integer ring $\...

0
votes

0
answers

252
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### Flatness over Hopf subalgebra

Let $A\subset B$ be flat Hopf algebras over a Dedekind ring $R$.
J. Moore has proved in the article
Compléments sur les algèbres de Hopf, John C. Moore, Séminaire Henri Cartan (1959-1960), Volume: ...

4
votes

2
answers

2k
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### Prime ideals in the ring of algebraic integers

Let $m(x) = x^n + a_{n-1}x^{n-1} + \dots + a_1 x+ a_0$, $a_i \in \mathbb{Z}$, be an irreducible polynomial over $\mathbb{Q}$ and $K = \mathbb{Q}[x] / {m(x)\mathbb{Q}[x]}$, so $K$ is an algebraic ...

5
votes

3
answers

1k
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### Orders of Number Fields

Let $K$ be a number field over $\mathbb{Q}$ of degree $n$, and $\mathcal{O} \subset \mathcal{O}_K$ an order.
$\textbf{Questions:}$
$\newcommand{\Spec}{\textrm{Spec }}$
$\newcommand{\cO}{\mathcal{O}}$
...

2
votes

2
answers

498
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### About the ring used in the definition of drinfeld modules. Why is that ring a dedekind domain?

Let $K/F$ be a function field with exact field of constants $F$ ($F$ is a finite field of characteristic $p$ prime). A prime in $K$ is a discrete valuation in $K$ containing $F$. It has a unique ...

1
vote

1
answer

429
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### The union of the totally split primes

Let $R$ be a Dedekind domain with quotient field $K$, let $L$ be a finite separable extension of $K$, and let $S$ be the integral closure of $R$ in $L$. If $\mathfrak{p}$ is a nonzero prime ideal of $...

0
votes

2
answers

561
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### What is the definition of the valuation of a fractional ideal?

I am reading Local fields and see Serre using $v_{\mathfrak{p}}(\mathfrak{a})$ where $\mathfrak{a}$ is a fractional ideal of the Dedekind domain $A$ and $v_{\mathfrak{p}}$ is the valuation associated ...

21
votes

4
answers

2k
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### Two questions about finiteness of ideal classes in abstract number rings

Let us say that an abstract number ring is an integral domain $R$ which is not a field, and which has the "finite norms" property: for any nonzero ideal $I$ of $R$, the quotient $R/I$ is finite.
(I ...

14
votes

4
answers

1k
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### Etale coverings of certain open subschemes in Spec O_K

Let $U$ be an open subscheme of $\textrm{Spec} \ \mathbf{Z}$. The complement of $U$ is a divisor $D$ of $\textrm{Spec} \ \mathbf{Z}$.
Q. Can we classify the etale coverings of $U$ of a given degree? ...

7
votes

1
answer

691
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### Do all Dedekind domains have the "Riemann-Roch property"?

Let $R$ be a Dedekind domain with fraction field $K$.
Say that a Dedekind domain $R$ has the Riemann-Roch property if: for every nonzero prime ideal $\mathfrak{p}$ of $R$, there exists an element $f ...