Polynomial Whose Values at the Integers are n-th Power of Integers in a Quadratic Field

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Janyarak Tongsomporn*
Vichian Laohakosol

Abstract

Let f(x1,x2,...,xk) gif.latex?\in gif.latex?\kappa [x1,x2,...,xk], where gif.latex?\kappa  is a quadratic field. We investigate the polynomial f (x1,x2,...,xkwhich becomes always an nth power of an quadratic integer using the technique of Kojima. It is shown that if f (gif.latex?\alpha1,gif.latex?\alpha2,...gif.latex?\alphak) is an nth power of an element in Ok , the ring of integers of gif.latex?\kappa, then f (x1,x2,...,xk)=(gif.latex?\varnothing(x1,x2,...xk))n,for somegif.latex?\varnothing (x1,x2,...,xkgif.latex?\inO[x1,x2,...xk].


Keywords: integer-valued polynomial, quadratic integer.


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

Article Details

Section
Original Research Articles

References

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