Eisenstein criterion

Theorem (Eisenstein criterion).

Let f be a primitive polynomial over a commutativePlanetmathPlanetmathPlanetmathPlanetmath unique factorization domainMathworldPlanetmath R, say


If R has an irreducible elementMathworldPlanetmath p such that

pam  0mn-1

then f is irreduciblePlanetmathPlanetmath.




where s>0 and t>0. Since a0=b0c0, we know that p divides one but not both of b0 and c0; suppose pc0. By hypothesisMathworldPlanetmath, not all the cm are divisible by p; let k be the smallest index such that pck. We have ak=b0ck+b1ck-1++bkc0. We also have pak, and p divides every summand except one on the right side, which yields a contradictionMathworldPlanetmathPlanetmath. QED ∎

Title Eisenstein criterion
Canonical name EisensteinCriterion
Date of creation 2013-03-22 12:16:32
Last modified on 2013-03-22 12:16:32
Owner Daume (40)
Last modified by Daume (40)
Numerical id 13
Author Daume (40)
Entry type Theorem
Classification msc 13A05
Synonym Eisenstein irreducibility criterion
Related topic GausssLemmaII
Related topic IrreduciblePolynomial2
Related topic Monic2
Related topic AlternativeProofThatSqrt2IsIrrational