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...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Journal of Algebra 633 (2023) 696–72
2023
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| Online Access: | http://hdl.handle.net/20.500.12493/1339 |
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| 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 F k , 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 for k, one of which only depends on the valuation of the
denominator of F and the size of the residue class field of R. |
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