Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains
Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (absolutely irreducibles) and irreducible elements where some power has a factorization different from the trivial one. In this paper...
Saved in:
| Main Authors: | , , |
|---|---|
| Published: |
2024
|
| Subjects: | |
| Online Access: | http://hdl.handle.net/20.500.12493/1947 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Rings of integer-valued polynomials are known to be atomic,
non-factorial rings furnishing examples for both irreducible
elements for which all powers factor uniquely (absolutely
irreducibles) and irreducible elements where some power has
a factorization different from the trivial one.
In this paper, we study irreducible polynomials F ∈ Int(R)
where R is a discrete valuation domain with finite residue field
and show that it is possible to explicitly determine a number
S ∈ N that reduces the absolute irreducibility of F to the
unique factorization of F S.
To this end, we establish a connection between the factors of
powers of F and the kernel of a certain linear map that we
associate to F. This connection yields a characterization of
absolute irreducibility in terms of this so-called fixed divisor
kernel. Given a non-trivial element v of this kernel, we
explicitly construct non-trivial factorizations of Fk, provided
that k ≥ L, where L depends on F as well as the choice of
v. We further show that this bound cannot be improved in
general. Additionally, we provide other (larger) lower bounds |
|---|