Irreducibility Criteria for Polynomials over Some Imaginary Quadratic Fields

摘要

Let K be an imaginary quadratic field whose ring of integers O_K is a Euclidean domain. In our earlier work, the following four results are proved. First, forβε O_K{0}, each element in O_K has a base β-expansion whose digits are bounded by certain constants. Secondly, if π = α_n β~n + α_(n?1)β~(n?1) + · · · + α1β+ α0 := f(β) is a base β-expansion of a prime in O_K, and the digits αn and α_(n?1) satisfy some natural restrictions, then the polynomial f(x) is irreducible over K. Thirdly, for Gaussian integers, similar base β-expansion but with digits belonging to a complete residue system modulo β is also valid. Lastly, irreducibility results similar to that in the second result continue to hold for Gaussian integers with base expansion as described in the third result. In this paper, we establish some generalizations of the above irreducibility results by considering ωπ (ωε O_K) instead of π.