What are Rings of Integer-Valued Polynomials?

Every integer is either even or odd, so we know that the polynomial f(x) = x(x− 1) 2 is integervalued on the integers, even though its coefficients are not in Z. Similarly, since every binomial coefficient ( k n ) is an integer, the polynomial ( x n ) = x(x− 1)...(x− n+ 1) n! must also be integervalued. These polynomials were used for polynomial interpolation as far back as the 17 century. Integer-valued polynomials did not become the subject of research on their own account until Pólya and Ostrowski considered the integer-valued polynomials on an algebraic number field K, that is the set Int(O) = {f(x) ∈ K[x] | f(O) ⊆ O}, where O is the ring of algebraic integers of K. Then in 1936, Thoralf Skolem was the first author to consider Int(Z) as a ring. Since then integervalued polynomial rings have been the subject of much study in commutative algebra.