@article{Phetnun2022, 
author = {Phitthayathon Phetnun and Narakorn R. Kanasri},
title = {Further irreducibility criteria for polynomials associated with the complete residue systems in any imaginary quadratic field},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {10},
pages = {18925-18947},
keywords = {imaginary quadratic field, ring of integers, complete residue system, irreducible element, irreducible polynomial},
url = {https://www.sciopen.com/article/10.3934/math.20221042},
doi = {10.3934/math.20221042},
abstract = {Let    K  =      Q    (      m    ) be an imaginary quadratic field with        O    K   its ring of integers. Let    π and    β be an irreducible element and a nonzero element, respectively, in        O    K  . In the authors' earlier work, it was proved for the cases,    m  ≢  1    (            m      o      d          4  ) and    m  ≡  1    (            m      o      d          4  ) that if    π  =      α    n        β    n    +      α          n      −      1            β          n      −      1        +  ⋯  +      α    1    β  +      α    0    =:  f  (  β  ), where    n  ≥  1,        α    n    ∈      O    K    ∖  {  0  },        α          0        ,  …  ,      α          n      −      1       belong to a complete residue system modulo    β, and the digits        α          n      −      1       and        α    n   satisfy certain restrictions, then the polynomial    f  (  x  ) is irreducible in        O    K    [  x  ]. In this paper, we extend these results by establishing further irreducibility criteria for polynomials in        O    K    [  x  ]. In addition, we provide elements of    β that can be applied to the new criteria but not to the previous ones.}
}