Ring of integer-valued polynomials over rational integers

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 $n$ can be interpolated using its values at any $n+1$ consecutive integers; moreover, the values at $n+1$ 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