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 OJDM  Vol.9 No.2 , April 2019
Irreducible Polynomials in Ζ[x] That Are Reducible Modulo All Primes
Shiv Gupta
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Abstract: The polynomial x4+1 is irreducible in Ζ[x] but is locally reducible, that is, it factors modulo p for all primes p. In this paper we investigate this phenomenon and prove that for any composite natural number N there are monic irreducible polynomials in Ζ[x] which are reducible modulo every prime.
Keywords: Irreducible Polynomial, Reducible Polynomial, Galois Theory
Cite this paper: Gupta, S. (2019) Irreducible Polynomials in Ζ[x] That Are Reducible Modulo All Primes. Open Journal of Discrete Mathematics, 9, 52-61. doi: 10.4236/ojdm.2019.92006.
References

[1]   Brandl, R. (1986) Integer Polynomials That Are Reducible Modulo All Primes. American Mathematical Monthly, 93, 286-288.
https://doi.org/10.1080/00029890.1986.11971807

[2]   Guralnick, R., Schacher, M.M. and Sonn, J. (2005) Irreducible Polynomials Which Are Locally Reducible Everywhere. Proceedings of the American Mathematical Society, 133, 3171-3177.
https://doi.org/10.1090/S0002-9939-05-07855-X

[3]   Safarevic, I.R. (1956) Construction of Fields of Algebraic Numbers with a Given Soluble Galois Group. Isv. Nauk. SSSR, 18, 274.

[4]   Schur, I. (1930) Gleichungen ohne Affeckt. Gesammelte Abhandlungen, Band III, No. 67, 191-197.

[5]   Serre, J.-P. (2008) Topics in Galois Theory. A K Peters, Ltd.

[6]   Erbach, D.W., Fisher, J. and McKay, J. (1979) Polynomials with PSL (2,7) as Galois Group. Journal of Number Theory, 11, 69-75.
https://doi.org/10.1016/0022-314X(79)90020-9

[7]   Lang, S. (1993) Algebra. 3rd Edition, Addison Wesley, Boston.

[8]   Wielandt, H. (1964) Finite Permutation Groups. Academic Press, New York.

[9]   Bernard Dominique, Email Communication.

[10]   Stevenhagen, P. and Lenstra, H.W. (1996) Chebotarev and His Density Theorem. The Mathematical Intelligencer, 18, 26-37.
https://doi.org/10.1007/BF03027290

 
 
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