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Irreducible polynomials which are locally reducible everywhere

Author(s): Robert Guralnick; Murray M. Schacher; Jack Sonn
Journal: Proc. Amer. Math. Soc. 133 (2005), 3171-3177.
MSC (2000): Primary 11R52, 11S25, 12F05, 12G05, 16K50
Posted: May 4, 2005
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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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R. Brandl, Integer polynomials that are reducible modulo all primes, Amer. Math. Monthly 93 (1986), 286-288. MR 0835298 (87f:12007)

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H. Kisilevsky and J. Sonn, On the $n$-torsion subgroup of the Brauer group of a number field, J. Th. Nombres de Bordeaux 15 (2003), 199-204. MR 2019011 (2004j:11142)

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G. Malle and B.H. Matzat, Inverse Galois Theory, Springer-Verlag, Berlin, 1999. MR 1711577 (2000k:12004)

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David Saltman, Generic Galois extensions and problems in field theory, Adv. Math. 43 (1982), 250-283. MR 0648801 (84a:13007)

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B. L. Van der Waerden, Die Seltenheit der Gleichungen mit Affekt, Math. Ann. 109 (1934), 13-16.

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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: 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
Posted: 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
Copyright of article: Copyright 2005, American Mathematical Society

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