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1answer
91 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 ...
3
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2answers
167 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 ...
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2answers
218 views

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 ...
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1answer
111 views

Intersection of powers of prime ideals

Let $R$ be a Noetherian ring. Let $(x)$ be a prime ideal such that $\bigcap_n (x)^n=0$. Then $R$ is a domain. Is this a known result? I heard its known as the Davis lemma. Can anyone give a ...
3
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1answer
153 views

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 ...
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1answer
151 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 ...
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2answers
402 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 ...
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0answers
159 views

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: ...
3
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1answer
461 views

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 $K = \mathbb{Q}(x) / {m(x)\mathbb{Q}(x)}$. K is an algebraic number ...
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3answers
492 views

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. $\textbf{Questions:}$ $\newcommand{\Spec}{\textrm{Spec }}$ $\newcommand{\cO}{\mathcal{O}}$ ...
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2answers
374 views

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 ...
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1answer
322 views

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
2answers
320 views

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 ...
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4answers
1k views

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 ...
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4answers
774 views

Etale coverings of certain open subschemes in Spec O_K

Although my number theory is really weak, I'm trying to understand the notion of etale coverings in this context. I think this could provide a very interesting point of view. Let $U$ be an open ...
3
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1answer
371 views

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 ...