# Every binomial polynomial is irreducible but not prime in the ring of integer-valued polynomials over rational integers

## Contents

## Statement

In the ring of integer-valued polynomials over rational integers, every binomial polynomial, i.e., every polynomial of the form:

is an irreducible element but not a prime element.

## Proof

### The binomial polynomial is irreducible

For this, we need a lemma: the largest number that we can guarantee divides the value of any polynomial of the form for all is .

With this lemma, we observe that if is expressed as a product of polynomials of smaller degree, each of them is of the form times a product of linear polynomials. In each of the cases, we get that is bounded from above by the factorial of the number of linear polynomials in that factor. Thus, the product of all the s is bounded by the product of the factorials, which is strictly less than , a contradiction.

### The binomial polynomial is not prime

For , we have:

.

This yields:

.

On the other hand, we have that since , does not divide any of the linear polynomials .

For , use that:

but does not divide , since the ratio, , is not an integer-valued polynomial.