## 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.

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 |
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Pages (from-to) | 231-249 |

Number of pages | 13 |

Journal | Journal of Algebra |

Volume | 528 |

DOIs | |

Publication status | Published - Jun 2019 |

## Fields of Expertise

- Information, Communication & Computing