Eisenstein criterion in terms of divisor theory

The below theorem generalises Eisenstein criterion of irreducibility from UFD’s to domains with divisor theoryMathworldPlanetmath.


Let  f(x):=a0+a1x++anxn  be a primitive polynomialMathworldPlanetmath over an integral domain 𝒪 with divisor theory (http://planetmath.org/DivisorTheory)  𝒪*𝔇.  If there is a prime divisor𝔭𝔇  such that

  • 𝔭a0,a1,,an-1,

  • 𝔭an,

  • 𝔭2a0,

then the polynomialMathworldPlanetmathPlanetmathPlanetmath is irreducible.

Proof.  Suppose that we have in 𝒪[x] the factorisation


with  s>0  and  t>0.  Because the principal divisor (a0), i.e. (b0)(c0) is divisible by the prime divisor 𝔭 and there is a unique factorisation in the monoid 𝔇, 𝔭 must divide (b0) or (c0) but, by 𝔭2(a0), not both of (b0) and (c0); suppose e.g. that 𝔭c0.  If 𝔭 would divide all the coefficients cj, then it would divide also the productMathworldPlanetmathPlanetmathPlanetmathbsct=an.  So, there is a certain smallest index k such that  pck.  Accordingly, in the sum b0ck+b1ck-1++bkc0, the prime divisor 𝔭 divides (http://planetmath.org/DivisibilityInRings) every summand except the first (see the definition of divisor theory (http://planetmath.org/DivisorTheory)); therefore it cannot divide the sum.  But the value of the sum is ak which by hypothesisMathworldPlanetmath is divisible by the prime divisor.  This contradictionMathworldPlanetmathPlanetmath shows that the polynomial f(x) is irreducible.

Title Eisenstein criterion in terms of divisor theory
Canonical name EisensteinCriterionInTermsOfDivisorTheory
Date of creation 2013-03-22 18:00:45
Last modified on 2013-03-22 18:00:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Theorem
Classification msc 13A05
Related topic DivisorTheory