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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A mild generalization of Eisenstein's criterion


Author: Steven H. Weintraub
Journal: Proc. Amer. Math. Soc. 141 (2013), 1159-1160
MSC (2010): Primary 12E05; Secondary 01A55
Published electronically: August 24, 2012
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Abstract: We state and prove a mild generalization of Eisenstein's Criterion for a polynomial to be irreducible, correcting an error that Eisenstein made himself.


References [Enhancements On Off] (What's this?)

  • 1. David A. Cox, Why Eisenstein proved the Eisenstein criterion and why Schönemann discovered it first, Normat 57 (2009), no. 2, 49–73, 96. MR 2572615 (2010j:01007)
  • 2. Eisenstein, F. G. M., Über die Irreductibilität und einige andere Eigenschaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt, J. reine angew. Math. 39 (1850), 160-179.
  • 3. Galois, E., Mémoire sur les conditions de résolubilité des équations par radicaux, J. Math. Pure Appl. 11 (1846), 381-444.
  • 4. Gauss, C. F., Disquisitiones Arithmeticae, Leipzig, 1801.
  • 5. Kronecker, L., Beweis dass für jede Primzahl $ p$ die Gleichung $ 1+x+\ldots +x^{p-1}=0$ irreductibel ist, J. reine angew. Math. 29 (1845), 280.
  • 6. Schönemann, T. Von denjenigen Moduln, welche Potenzen von Primzahlen sind, J. reine angew. Math. 32 (1846), 93-105.

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Additional Information

Steven H. Weintraub
Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvannia 18015-3174
Email: shw2@lehigh.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-10880-9
PII: S 0002-9939(2012)10880-9
Keywords: Eisenstein’s criterion, irreducibility
Received by editor(s): November 3, 2010
Received by editor(s) in revised form: August 12, 2011
Published electronically: August 24, 2012
Communicated by: Ken Ono
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.