Current Applied Science and Technology, Volume 17, Issue 1, Pages 13-21 , 01/01/2017

Polynomial whose values at the integers are n-th power of integers in a quadratic field

Janyarak Tongsomporn, Vichian Laohakosol

Abstract

Let f (x <inf>1</inf> , x <inf>2</inf> ,…,x <inf>k</inf> ) ∈K [x <inf>1</inf> , x <inf>2</inf> ,…,x <inf>k</inf> ], where K is a quadratic field. We investigate the polynomial f (x <inf>1</inf> ,x <inf>2</inf> ,…,x <inf>k</inf> ) which becomes always an n <sup>th</sup> power of an quadratic integer using the technique of Kojima. It is shown that if f (α <inf>1</inf> ,α <inf>2</inf> ,…,α <inf>k</inf> )is an n <sup>th</sup> power of an element in O <inf>K</inf> , the ring of integers of K, then f(x <inf>1</inf> ,x <inf>2</inf> ,…, x <inf>k</inf> ) = (ϕ (x <inf>1</inf> , x <inf>2</inf> ,…, x <inf>k</inf> )) <sup>n</sup> , for some ϕ(x <inf>1</inf> ,x <inf>2</inf> ,…,x <inf>k</inf> )∈O <inf>K</inf> [x,x <inf>1 2</inf> ,…,x <inf>k</inf> ].

Document Type

Article

Source Type

Journal

Keywords

Integer-valued polynomialQuadratic integer

ASJC Subject Area

Agricultural and Biological Sciences : Agronomy and Crop ScienceAgricultural and Biological Sciences : Agricultural and Biological Sciences (miscellaneous)Biochemistry, Genetics and Molecular Biology : BiotechnologyEnvironmental Science : Environmental Engineering

Funding Agency

Kasetsart University

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Tongsomporn, J., & Laohakosol, V. (2017). Polynomial whose values at the integers are n-th power of integers in a quadratic field. Current Applied Science and Technology, 17(1) 13-21.

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