Irreducible polynomials which are locally reducible everywhere
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- by Robert Guralnick, Murray M. Schacher and Jack Sonn PDF
- Proc. Amer. Math. Soc. 133 (2005), 3171-3177 Request permission
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.References
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Additional Information
- Robert Guralnick
- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
- MR Author ID: 78455
- 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
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 3171-3177
- MSC (2000): Primary 11R52, 11S25, 12F05, 12G05, 16K50
- DOI: https://doi.org/10.1090/S0002-9939-05-07855-X
- MathSciNet review: 2160178