Base-\beta(C) Representations and Generalizations of Irreducibility Criteria for Polynomials over any Imaginary Quadratic Fields
DOI:
https://doi.org/10.29020/nybg.ejpam.v18i1.5709Keywords:
Imaginary quadratic field, Ring of integers, Complete residue system, Prime element, Irreducible polynomialAbstract
Let K be an imaginary quadratic field with the ring of integers O_K. In the authors' earlier work, the so-called base-\beta(C) representation for nonzero elements of O_K was determined, where C is a complete residue system modulo \beta. Using such a representation, irreducibility criteria for polynomials in O_K[x] were established. In this paper, we provide the explicit shapes of all base-\beta(C) representations for nonzero elements of O_K. Generalizations of such irreducibility criteria for polynomials in O_K[x] under a certain condition are also established.
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