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 $\operatorname{Int}(D) = \{f\in K[x] \mid f(D) \subseteq D\}$ 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 $\operatorname{Int}(D)$ which has exactly $n$ essentially different factorizations into irreducibles in $\operatorname{Int}(D)$, 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 $\operatorname{Int}(D)$ to a block monoid.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2017
- DOI:
- 10.48550/arXiv.1710.06783
- arXiv:
- arXiv:1710.06783
- Bibcode:
- 2017arXiv171006783F
- Keywords:
-
- Mathematics - Commutative Algebra;
- 13A05;
- 13B25;
- 13F20;
- 11R04;
- 11C08
- E-Print:
- doi:10.1016/j.jalgebra.2019.02.040