Factorization behaviour of the binomials and their powers in the ring of integer-valueds

Absolute irreducibility of the binomial polynomials

We investigate the factorization behaviour of the binomial polynomials and their powers in the ring of integer-valued polynomials.We show that the binomial polynomials are absolutely irreducible in that is, factors uniquely into irreducible elements in for all 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 the binomial polynomials.A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof : if and \ldots, are composite integers, then there exists a prime number that divides one of these integers.