Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields

Sophie Frisch, Sarah Nakato, Roswitha Rissner

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Let $D$ be a Dedekind domain with infinitely many maximal ideals,
all of finite index, and $K$ its quotient field. Let
$\Int(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of
integer-valued polynomials on $D$.

Given any finite multiset $\{k_1, \ldots, k_n\}$ of integers greater
than $1$, we construct a polynomial in $\Int(D)$ which has exactly
$n$ essentially different factorizations into irreducibles in
$\Int(D)$, the lengths of these factorizations being $k_1$, \ldots,
$k_n$. We also show that there is no transfer homomorphism from the
multiplicative monoid of $\Int(D)$ to a block monoid.
Original languageEnglish
Pages (from-to)231-249
Number of pages13
JournalJournal of algebra
Volume528
DOIs
Publication statusPublished - Jun 2019

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Integer-valued Polynomials
Dedekind Domain
Monoid
Factorization
Maximal Ideal
Multiset
Homomorphism
Multiplicative
Quotient
Ring
Polynomial
Integer

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Cite this

Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields. / Frisch, Sophie; Nakato, Sarah; Rissner, Roswitha.

In: Journal of algebra, Vol. 528, 06.2019, p. 231-249.

Research output: Contribution to journalArticleResearchpeer-review

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AB - Let $D$ be a Dedekind domain with infinitely many maximal ideals, all of finite index, and $K$ its quotient field. Let $\Int(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ be the ring of integer-valued polynomials on $D$. Given any finite multiset $\{k_1, \ldots, k_n\}$ of integers greater than $1$, we construct a polynomial in $\Int(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\Int(D)$, the lengths of these factorizations being $k_1$, \ldots, $k_n$. We also show that there is no transfer homomorphism from the multiplicative monoid of $\Int(D)$ to a block monoid.

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