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

 

Irreducible polynomials which are locally reducible everywhere


Authors: Robert Guralnick, Murray M. Schacher and Jack Sonn
Journal: Proc. Amer. Math. Soc. 133 (2005), 3171-3177
MSC (2000): Primary 11R52, 11S25, 12F05, 12G05, 16K50
Published electronically: May 4, 2005
MathSciNet review: 2160178
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Abstract | References | Similar Articles | Additional Information

Abstract: For any positive integer $n$, there exist polynomials $f(x)\in \mathbb{Z} [x]$of degree $n$ which are irreducible over $\mathbb{Q} $ and reducible over $\mathbb{Q} _{p}$ for all primes $p$ if and only if $n$ is composite. In fact, this result holds over arbitrary global fields.


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Additional Information

Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: guralnic@usc.edu

Murray M. Schacher
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90024
Email: mms@math.ucla.edu

Jack Sonn
Affiliation: Department of Mathematics, Technion, 32000 Haifa, Israel
Email: sonn@math.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07855-X
PII: S 0002-9939(05)07855-X
Received by editor(s): April 3, 2004
Received by editor(s) in revised form: June 17, 2004
Published electronically: May 4, 2005
Additional Notes: The first author was partially supported by NSF Grant DMS 0140578. The research of the third author was supported by Technion V.P.R. Fund–S. and N. Grand Research Fund
Communicated by: Martin Lorenz
Article copyright: © Copyright 2005 American Mathematical Society