### Abstract

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 |
---|---|

Pages (from-to) | 231-249 |

Number of pages | 13 |

Journal | Journal of algebra |

Volume | 528 |

DOIs | |

Publication status | Published - Jun 2019 |

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### Fields of Expertise

- Information, Communication & Computing

### Cite this

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

Research output: Contribution to journal › Article › Research › peer-review

}

TY - JOUR

T1 - Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields

AU - Frisch, Sophie

AU - Nakato, Sarah

AU - Rissner, Roswitha

PY - 2019/6

Y1 - 2019/6

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

SP - 231

EP - 249

JO - Journal of algebra

JF - Journal of algebra

SN - 0021-8693

ER -