# Ring of integer-valued polynomials over rational integers

From Commalg

## Definition

### Symbol-free definition

The **ring of integer-valued polynomials over rational integers** is defined in the following equivalent ways:

- It is the ring of integer-valued polynomials over the ring of rational integers.
- It is the ring generated by binomial polynomials over the ring of rational integers.

### Equivalence of definitions

*For full proof, refer: Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials*

## Facts

- Ring of integer-valued polynomials over rational integers is an interpolation domain: An integer-valued polynomial of degree can be interpolated using its values at any consecutive integers; moreover, the values at consecutive integers can be any tuple whatsoever.
- Ring of integer-valued polynomials over rational integers is not Noetherian
- Ring of integer-valued polynomials over rational integers is not a UFD