TY - JOUR
T1 - Absolute irreducibility of the binomial polynomials
AU - Windisch, Daniel
PY - 2021/7/15
Y1 - 2021/7/15
N2 - In this paper we investigate the factorization behaviour of the binomial polynomials [Formula presented] and their powers in the ring of integer-valued polynomials Int(Z). While it is well-known that the binomial polynomials are irreducible elements in Int(Z), the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int(Z), that is, (xn)
m factors uniquely into irreducible elements in Int(Z) for all m∈N. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n>10 and n, n−1, …, n−(k−1) are composite integers, then there exists a prime number p>2k that divides one of these integers.
AB - In this paper we investigate the factorization behaviour of the binomial polynomials [Formula presented] and their powers in the ring of integer-valued polynomials Int(Z). While it is well-known that the binomial polynomials are irreducible elements in Int(Z), the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in Int(Z), that is, (xn)
m factors uniquely into irreducible elements in Int(Z) for all m∈N. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of n. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If n>10 and n, n−1, …, n−(k−1) are composite integers, then there exists a prime number p>2k that divides one of these integers.
KW - Absolute irreducibility
KW - Binomial polynomials
KW - Factorization theory
KW - Integer-valued polynomials
UR - http://www.scopus.com/inward/record.url?scp=85103054492&partnerID=8YFLogxK
U2 - 10.1016/j.jalgebra.2021.03.007
DO - 10.1016/j.jalgebra.2021.03.007
M3 - Article
SN - 0021-8693
VL - 578
SP - 92
EP - 114
JO - Journal of Algebra
JF - Journal of Algebra
ER -