Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-6

Characterization of irreducible polynomials over a special principal ideal ring

Brahim Boudine

Received December 5, 2021. Published online September 8, 2022.

Abstract: A commutative ring $R$ with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length $e$ is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length $2$. Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length $e$.

Keywords: polynomial; irreducibility; commutative principal ideal ring

Classification MSC: 13F20, 13B25

DOI: 10.21136/MB.2022.0187-21

PDF available at: Institute of Mathematics CAS

References:
[1] G. Azumaya: On maximally central algebras. Nagoya Math. J. 2 (1951), 119-150. DOI 10.1017/S0027763000010114 | MR 0040287 | Zbl 0045.01103
[2] E. R. Berlekamp: Algebraic Coding Theory. McGraw-Hill, New York (1968). DOI 10.1142/9407 | MR 0238597 | Zbl 0988.94521
[3] W. C. Brown: Matrices Over Commutative Rings. Pure and Applied Mathematics 169. Marcel Dekker, New York (1993). MR 1200234 | Zbl 0782.15001
[4] M. E. Charkani, B. Boudine: On the integral ideals of $R [X]$ when $R$ is a special principal ideal ring. São Paulo J. Math. Sci. 14 (2020), 698-702. DOI 10.1007/s40863-020-00177-1 | MR 4173484 | Zbl 1451.13028
[5] B. Chor, R. L. Rivest: A knapsack-type public key cryptosystem based on arithmetic in finite fields. IEEE Trans. Inf. Theory 34 (1988), 901-909. DOI 10.1109/18.21214 | MR 0982801 | Zbl 0664.94011
[6] D. Eisenbud: Commutative Algebra: With a View Toward Algebraic Geometry. Graduate Texts in Mathematics 150. Springer, Berlin 1995. DOI 10.1007/978-1-4612-5350-1 | MR 1322960 | Zbl 0819.13001
[7] D. Hachenberger, D. Jungnickel: Irreducible polynomials over finite fields. Topics in Galois Fields Algorithms and Computation in Mathematics 29. Springer, Cham (2020), 197-239. DOI 10.1007/978-3-030-60806-4_5
[8] M. O. Rabin: Probabilistic algorithms in finite fields. SIAM J. Comput. 9 (1980), 273-280. DOI 10.1137/0209024 | MR 0568814 | Zbl 0461.12012
[9] C. Rotthaus: Excellent rings, Henselian rings and the approximation property. Rocky Mt. J. Math. 27 (1997), 317-334. DOI 10.1216/rmjm/1181071964 | MR 1453106 | Zbl 0881.13009

Affiliations: Brahim Boudine, Faculty of Sciences Dhar El Mahraz, University Sidi Mohamed Ben Abdellah, Fez, Morocco, e-mail: brahimboudine.bb@gmail.com

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