A. Cohn’s irreducibility criterion
Theorem.
Assume $n\mathrm{\ge}\mathrm{2}$ is an integer and that $P$ is a polynomial^{} with coefficients in $\mathrm{\{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}n\mathrm{-}\mathrm{1}\mathrm{\}}$. If $P\mathit{}\mathrm{(}n\mathrm{)}$ is prime then $P\mathit{}\mathrm{(}x\mathrm{)}$ is irreducible (http://planetmath.org/IrreduciblePolynomial2) in $\mathrm{Z}\mathit{}\mathrm{[}x\mathrm{]}$.
A proof is given in [MRM].
A. Cohn [PZ] proved this theorem for the case $n=10$.
This special case of the above theorem is sketched as problem 128, Part VIII, in [PZ].
References
- PZ George PÃÂ³lya, Gabor Szego, Problems and Theorems in Analysis II, Classics in Mathematics 1998.
- MRM M. Ram Murty, Prime Numbers and Irreducible Polynomials^{}, American Mathematical Monthly, vol. 109, (2002), 452-458.
Title | A. Cohn’s irreducibility criterion |
---|---|
Canonical name | ACohnsIrreducibilityCriterion |
Date of creation | 2013-03-22 14:37:02 |
Last modified on | 2013-03-22 14:37:02 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 17 |
Author | Mathprof (13753) |
Entry type | Theorem |
Classification | msc 11C08 |