# 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 language English 231-249 13 Journal of algebra 528 https://doi.org/10.1016/j.jalgebra.2019.02.040 Published - Jun 2019

### Fingerprint

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

### Fields of Expertise

• Information, Communication & Computing

### Cite this

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

Research output: Contribution to journalArticleResearchpeer-review

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title = "Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields",
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.",
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AU - Frisch, Sophie

AU - Nakato, Sarah

AU - Rissner, Roswitha

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N2 - 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.

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.

U2 - 10.1016/j.jalgebra.2019.02.040

DO - 10.1016/j.jalgebra.2019.02.040

M3 - Article

VL - 528

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EP - 249

JO - Journal of algebra

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SN - 0021-8693

ER -