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A note on Perron’s irreducibility criterion

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Abstract

In this note, we generalize the classical irreducibility criterion of Perron and obtain numerous other new irreducibility criteria for polynomials having integer coefficients.

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References

  1. Bonciocat, N.C.: On an irreducibility criterion of Perron for multivariate polynomials. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101), 213–217 (2010)

    MathSciNet  MATH  Google Scholar 

  2. Bonciocat, A.I., Bonciocat, N.C.: Some classes of irreducible polynomials. Acta Arith. 123, 349–360 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonciocat, A.I., Bonciocat, N.C.: On the irreducibility of polynomials with leading coefficient divisible by a large prime power. Amer. Math. Mon. 116(8), 743–745 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonciocat, A.I., Bonciocat, N.C., Zaharescu, A.: On the irreducibility of polynomials that take a prime power value. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54(102), 41–54 (2011)

  5. Dumas, G.: Sur quelques cas d’irréductibilité des polynomes à coefficients rationnels. J. Math. Pure Appl. 2, 191–258 (1906)

  6. Eisenstein, G.: Über die Irreduzibilität und einige andere Eigenschaften der Gleichungen. J. Reine Angew. Math. 39, 160–179 (1850)

    MathSciNet  MATH  Google Scholar 

  7. Lipka, S.: Über die Irreduzibilität von Polynomen. Math. Ann. 118, 235–245 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  8. Panitopol, L., Ştefănescu, D.: Some criteria for irreducibility of polynomials. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 29, 69–74 (1985)

    MathSciNet  MATH  Google Scholar 

  9. Perron, O.: Neue Kriterien für die Irreduzibilität algebraischer Gleichungen. J. Reine Angew. Math. 132, 288–307 (1907)

    MathSciNet  MATH  Google Scholar 

  10. Schönemann, T.: Von denjenigen Moduln, welche Potenzen von Primzahlen sind. J. Reine Angew. Math. 32, 93–105 (1846)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are indebted to the anonymous referee for valuable suggestions in improving the article. The second author is thankful to the Council of Scientific and Industrial Research (CSIR) for providing her Junior Research Fellowship (JRF) wide grant no. CSIRAWARD/JRF-NET2022/11769 for carrying out the present research.

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  1. Department of Mathematics, Guru Nanak Dev University, Amritsar, 143005, India

    Jitender Singh & Rishu Garg

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  1. Jitender Singh
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  2. Rishu Garg
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Correspondence to Jitender Singh.

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Singh, J., Garg, R. A note on Perron’s irreducibility criterion. Arch. Math. 121, 33–38 (2023). https://doi.org/10.1007/s00013-023-01873-y

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  • DOI: https://doi.org/10.1007/s00013-023-01873-y

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