Base-𝛽 Representations and Irreducibility of Polynomials over Any Imaginary Quadratic Field
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Abstract
Let 𝐾 be an imaginary quadratic field whose ring of integers 𝑂𝐾 is a Euclidean domain. In the earlier work, the so-called base-𝛽 representation for nonzero elements of 𝑂𝐾 was constructed and the irreducibility criterion for polynomials in 𝑂𝐾 [𝑥] was established,
namely if 𝜋 = 𝛼𝑛 𝛽 𝑛 + 𝛼𝑛−1 𝛽 𝑛−1 + · · · + 𝛼1 𝛽 + 𝛼0 =: 𝑓 (𝛽) is a base-𝛽 representation of a prime element 𝜋 ∈ 𝑂𝐾 and the digits 𝛼𝑛−1 and 𝛼𝑛 satisfy some natural restrictions, then the polynomial 𝑓 (𝑥) is irreducible in 𝑂𝐾 [𝑥]. A generalization of this criterion was also verified by considering 𝜔𝜋 (𝜔 ∈ 𝑂𝐾 \{0}) instead of 𝜋. In this paper, we extend these results to any imaginary quadratic field 𝐾.
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