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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.

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. Reference: Wikipedia.

It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition.

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Scaling of Fractional ideals

For fractional ideals of a Dedekind Domain, are each of the elements that generate the ideal (ie. form the basis of the lattice associated with the ideal) always scaled by the same amount? That is to say, scaled by the same element from the field…
Mike Four
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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 understand about the theorem Let $[K:Q]=n$ with the ring…
bgj123
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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 $A$), but if we contract $\mathfrak b$ to $\mathfrak…
Sha Vuklia
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Prime ideal in Dedekind domain

Let $\mathcal{o}$ be a Dedekind domain, $K$ its field of fractions, $L$ a finite separable field extension of $K$, and $\mathcal{O}$ the integral closure of $\mathcal{o}$ in $L$. When $\mathfrak{p}$ is a prime ideal in the Dedekind domain…
yhsai
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