# 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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### Tor-dimension of $A/(a)\otimes_k B$, where $A$ and $B$ are Dedekind domains

Let $A$ and $B$ be two Dedekind domains which contain a field $k$ which is algebraically closed in both $A$ and $B$. Let $a$ be a non zero element in $A$. What is the Tor-dimension of $A\otimes_k B$? ...
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### Prime of order is regular iff its decomposition in the normalization is trivial.

It's from a statement in Algebraic Number Theory by Neukirch, page 92. Example 5 "One can show..." Let ${o}$ be a one-dimensional noetherian integral domain and $\tilde{o}$ be its ...
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### $M$ $\mathscr{R}$-submodule complete lattice $\iff$ $aL \subset M \subset a^{-1}L$, $a \in \mathscr{R}$

My question may be nestled in the sense that there might be confusion about several notions, please let me know if it is the case. In "The Arithmetic of Hyperbolic 3-manifolds" by Maclachlan ...
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### Coordinate Rings and Dedekind Domains

I am relatively new to ring theory, though I have an idea of what things kind of are. I have been learning about Dedekind domains, which are integral domains which are Noetherian, integrally closed ...
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### Theorem $17$, Chapter $3$ (Marcus' Number Fields): Every ideal in a Dedekind domain is generated by at most two elements

I understand that similar questions have been asked before, but I am looking for an explanation of certain steps in Marcus' proof of the same, in Theorem $17$, Chapter $3$ of Number Fields. I shall ...
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### Why is the class group torsion?

While wandering through math stack exchange I found an interesting question, namely this one: Why are the algebraic integers a Bezout Domain? (Found here: Is there an elementary way to prove that the ...
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### Find all ideals contains power of prime ideals

Let $R$ be a Dedekind domain. Let $P$ be a prime ideal of $R$. Can we figure out all ideals which satisfies $P^n⊂I⊂R$ ? Factorlization of $I$ into maximal ideals should help, but I don't exactly what ...
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### the sum of the products of ramification degrees and relative degrees

I am reading Algebraic Number Fields by Gerald Janusz and I get confused about the part in the picture below. Consider two Dedekind domains $R\subset R'$ with quotient fields $K\subset L$. Let $p$ be ...
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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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### 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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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 $\mathfrak{... 1 vote 0 answers 90 views ### Doubt regarding the two definitions of Norm of an ideal in Algebraic Number theory Let$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$, we ... 0 votes 1 answer 31 views ### Arithmetic structure including both unique factorization and Dedekind domains Has an algebraic arithmetic structure been defined on integral domains, which would include both Dedekind rings and unique factorization domain with respect to the arithmetic properties, and more ... 1 vote 0 answers 79 views ### Dedekind's Theorem Use Dedekind’s Theorem to factorise the following principal ideals in the ring of integers of the following fields. a)$Q(√3): ⟨2⟩,⟨3⟩,⟨5⟩,⟨30⟩$b)$Q( ^3√2): ⟨7⟩, ⟨29⟩, ⟨31⟩$Here is what I ... 1 vote 0 answers 89 views ### Resolutions over Dedekind Domains I want to prove that if$R$is a Dedekind Domain, then$\operatorname{Ext}_R^n(M,N)=0$, for$n \geq 2$. Then, I have a question: There are any properties about projective or injective resolutions over ... 1 vote 1 answer 44 views ### Is discrete valuation field a fraction field of some Dedekind domain? Is discrete valuation field a fraction field of some Dedekind domain ? Let$K$be a discrete valuation field, does there exist some Dedekind domain$R$such that$\operatorname{Frac}(R)$＝$K$？ If ... 3 votes 0 answers 296 views ### Proposition 8.3, Neukirch- Algebraic Number Theory Let$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$be a primitive ... 1 vote 2 answers 124 views ### Examples of modules that$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} \... 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 ...