Base-𝛽 Representations and Irreducibility of Polynomials over Any Imaginary Quadratic Field
Keywords:
Gauss’s lemma, Imaginary quadratic field, Irreducible element, Irreducible polynomial, Ring of integersAbstract
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.
References
Pólya G, Szegö G. Problems and theorems in analysis. New York: SpringerVerlag; 1976.
Brillhart J, Filaseta M, Odlyzko A. On an irreducibility theorem of A. Cohn. Canad. J. Math. 1981;33(5):1055-9.
Murty MR. Prime numbers and irreducible polynomials. Amer. Math. Monthly 2002;109(5):452-8.
Filaseta M. A further generalization of an irreducibility theorem of A. Cohn. Canad. J. Math. 1982;34(6):1390-5.
Alaca S, Williams KS. Introductory algebraic number theory. Cambridge: Cambridge University Press; 2004.
Nicholson WK. Introduction to abstract algebra. 3rd ed. New Jersey: Wiley; 2007.
Rosen KH. Elementary number theory and its applications. 5th ed. New York: Addison-Wesley; 2005.
Singthongla P, Kanasri NR, Laohakosol V. Prime elements and irreducible polynomials over some imaginary quadratic fields. Kyungpook Math. 2017;57(4):581-600.
Kanasri NR, Singthongla P, Laohakosol V. Irreducibility criteria for polynomials over some imaginary quadratic fields. Southeast Asian Bull. Math. 2019;43(1):367-76.
Tadee S, Laohakosol V, Damkaew S. Explicit complete residue systems in a general quadratic field. Divulg. Mat. 2017;18(2):1-17.
Phetnun P, Kanasri NR, Singthongla P. On the irreducibility of polynomials associated with the complete residue systems in any imaginary quadratic fields. Int. J. Math. Math. Sci. 2021;2021:17 pages.
Phetnun P, Kanasri NR. Further irreducibility criteria for polynomials associated with the complete residue systems in any imaginary quadratic field. AIMS Math. 2022;7(10):18925-47.
Andreescu T, Andrica D, Cucurezeanu I. An introduction to Diophantine equations: a problem-based approach. New York: Birkhäuser; 2010.
Malik DS, Mordeson JN, Sen MK. Fundamentals of abstract algebra. New York: McGraw-Hill; 1997.
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