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Original Articles

Schönemann–Eisenstein–Dumas-Type Irreducibility Conditions that Use Arbitrarily Many Prime Numbers

Pages 3102-3122 | Received 03 Feb 2014, Published online: 04 Jun 2015

REFERENCES

  • Bauer , M. ( 1905 ). Verallgemeinerung eines Satzes von Schönemann . J. Reine Angew. Math. 128 : 87 – 89 .
  • Bishnoi , A. , Khanduja , S. K. ( 2010 ). On Eisenstein-Dumas and Generalized Schönemann polynomials . Comm. Algebra 38 : 3163 – 3173 .
  • Brown , R. ( 2008 ). Roots of generalized Schönemann polynomials in Henselian extension fields . Indian J. Pure Appl. Math. 39 ( 5 ): 403 – 410 .
  • Bush , M. R. , Hajir , F. ( 2008 ). An irreducibility lemma . J. Ramanujan Math. Soc. 23 ( 1 ): 33 – 41 .
  • Coleman , R. F. ( 1987 ). On the Galois groups of the exponential Taylor polynomials . Enseign. Math. (2) 33 ( 3–4 ): 183 – 189 .
  • Cox , D. A. ( 2011 ). Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first . Amer. Math. Monthly 118 ( 1 ): 3 – 21 .
  • Dumas , G. ( 1906 ). Sur quelques cas d'irreductibilité des polynômes á coefficients rationnels . Journal de Math. Pure et Appl. 2 : 191 – 258 .
  • Eisenstein , G. ( 1850 ). Über die Irreductibilität und einige andere Eigenschaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt . J. Reine Angew. Math. 39 : 160 – 182 .
  • Filaseta , M. ( 1995 ). On the irreducibility of almost all Bessel Polynomials . Acta Math. 174 ( 2 ): 383 – 397 .
  • Filaseta , M. ( 1996 ). A generalization of an irreducibility theorem of I. Schur . In: Berndt , B. C. , Diamond , H. G. , Hildebrand , A. J. , eds. Analytic Number Theory, Proc. Conf. in Honor of Heini Halberstam , Vol. 1 . Boston : Birkhäuser , pp. 371 – 395 .
  • Filaseta , M. , Kidd , T. Trifonov , O. ( 2012 ). Laguerre polynomials with Galois group A m for each m . J. Number Theory 132 : 776 – 805 .
  • Filaseta , M. , Finch , C. , Leidy , J. R. (2005). T.N. Shorey's influence in the theory of irreducible polynomials. In: Saradha , N. , ed. Diophantine Equations . New Delhi : Narosa Publ. House, pp. 77–102.
  • Filaseta , M. , Lam , T.-Y. ( 2002 ). On the irreducibility of the Generalized Laguerre polynomials . Acta Arith. 105 ( 2 ): 177 – 182 .
  • Filaseta , M. , Trifonov , O. ( 2002 ). The irreducibility of the Bessel Polynomials . J. Reine Angew. Math. 550 : 125 – 140 .
  • Filaseta , M. , Williams , R. L. Jr. ( 2003 ). On the irreducibility of a certain class of Laguerre polynomials . J. Number Theory 100 ( 2 ): 229 – 250 .
  • Greve , C. , Pauli , S. ( 2012 ). Ramification polygons, splitting fields, and Galois groups of Eisenstein polynomials . Int. J. Number Theory 8 ( 6 ): 1401 – 1424 .
  • Hajir , F. ( 1995 ). Some A n -extensions obtained from generalized Laguerre polynomials . J. Number Theory 50 : 206 – 212 .
  • Hajir , F. ( 2005 ). On the Galois group of generalized Laguerre polynomials . J. Théor. Nombres Bordeaux 17 : 517 – 525 .
  • Hajir , F. ( 2009 ). Algebraic properties of a family of generalized Laguerre polynomials . Canad J. Math. 61 ( 3 ): 583 – 603 .
  • Hajir , F. , Wong , S. ( 2006 ). Specializations of one-parameter families of polynomials . An. Inst. Fourier 56 ( 4 ): 1127 – 1163 .
  • Khanduja , S. K. , Khassa , R. ( 2011 ). A generalization of Eisenstein-Schönemann irreducibility criterion . Manuscripta Math. 134 : 215 – 224 .
  • Khanduja , S. K. , Saha , J. ( 1997 ). On a generalization of Eisenstein's irreducibility criterion . Mathematika 44 : 37 – 41 .
  • Königsberger , L. ( 1895 ). Über den Eisensteinschen Satz von der Irreduzibilität algebraischer Gleichungen . J. Reine Angew. Math. 115 : 53 – 78 .
  • Kurschak , J. ( 1923 ). Irreduzible Formen . J. Reine Angew. Math. 152 : 180 – 191 .
  • MacLane , S. ( 1938 ). The Schönemann-Eisenstein irreducibility criteria in terms of prime ideals . Trans. Amer Math. Soc. 43 : 226 – 239 .
  • Mott , J. ( 1995 ). Eisenstein-type irreducibility criteria . In: Zero-dimensional Commutative Rings (Knoxville, TN, 1994) . Lecture Notes in Pure and Appl. Math. , Vol. 171 , New York : Dekker , pp. 307 – 329 .
  • Netto , E. ( 1896 ). Über die Irreduzibilität ganzahligen ganzer Funktionen . Math. Ann. 48 : 81 – 88 .
  • Ore , O. ( 1923 ). Zur Theorie der IrreduzibilitätsKriterien . Math. Zeit. 18 : 278 – 288 .
  • Ore , O. ( 1924 ). Zur Theorie der Eisensteinschen Gleichungen . Math. Zeit. 20 : 267 – 279 .
  • Ore , O. ( 1924 ). Zur Theorie der Algebraischen Körper . Acta Math. 44 : 219 – 314 .
  • Panaitopol , L. , Ştefănescu , D. ( 1988 ). Factorization of the Schönemann polynomials . Bull. Math. Soc. Sci. Math. Roumanie (N. S.) 32 ( 80)(3 ): 259 – 262 .
  • Perron , O. ( 1905 ). Uber eine Anwendung der Idealtheorie auf die Frage nach der Irreduzibilität algebraischen Gleichengen . Math. Ann. 60 : 448 – 458 .
  • Prasolov , V. V. ( 2010 ). Polynomials (Translated from the 2001 Russian second edition by Dimitry Leites) . In: Algorithms and Computation in Mathematics , 11 . Berlin : Springer .
  • Schönemann , T. ( 1846 ). Von denjenigen Moduln, welche Potenzen von Primzahlen sind . J. Reine Angew. Math. 32 : 93 – 105 .
  • Sell , E. A. ( 2004 ). On a certain family of Generalized Laguerre Polynomials . J. Number Theory 107 ( 2 ): 266 – 281 .
  • Weintraub , S. H. ( 2013 ). A mild generalization of Eisenstein's criterion . Proc. Amer. Math. Soc. 141 ( 4 ): 1159 – 1160 .
  • Communicated by T. Albu.

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