# 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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### 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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### Prime ideal factorization in non-monogenic ring of integers ($K=\mathbb{Q}(\sqrt{24})$)

Consider the number field $K=\mathbb{Q}(\sqrt{24})$. Then one can show that $\mathcal{O}_{K}=\mathbb{Z}[\alpha,\beta]$ where $\alpha=\sqrt{24}$ 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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### 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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### 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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