Polynomial Whose Values at the Integers Are N-Th Power of Integers in a Quadratic Field

Let f(x_1,x2,...,x_k)   [x₁,x₂,...,x_k], where   is a quadratic field. We investigate the polynomial f (x₁,x_2,...,x_k) which becomes always an n^th power of an quadratic integer using the technique of Kojima. It is shown that if f (₁,₂,..._k) is an n^_th power of an element in O_k , the ring of integers of , then f (x₁,x₂,...,x_k)=((x₁,x₂,...x_k))ⁿ,for some (x₁,x₂,...,x_k) O_k [x₁,x₂,...x_k].

Keywords: integer-valued polynomial, quadratic integer.

^*Corresponding author:       E-mail: janyarak.to@wu.ac.th