Eisenstein criterion in terms of divisor theory
The below theorem generalises Eisenstein criterion of irreducibility from UFD’s to domains with divisor theory^{}.
Theorem.
Let $f(x):={a}_{0}+{a}_{1}x+\mathrm{\dots}+{a}_{n}{x}^{n}$ be a primitive polynomial^{} over an integral domain $\mathcal{O}$ with divisor theory (http://planetmath.org/DivisorTheory) ${\mathcal{O}}^{*}\to \U0001d507$. If there is a prime divisor $\U0001d52d\in \U0001d507$ such that

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$\U0001d52d\mid {a}_{0},{a}_{1},\mathrm{\dots},{a}_{n1},$

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$\U0001d52d\nmid {a}_{n},$

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${\U0001d52d}^{2}\nmid {a}_{0},$
then the polynomial^{} is irreducible.
Proof. Suppose that we have in $\mathcal{O}[x]$ the factorisation
$$f(x)=({b}_{0}+{b}_{1}x+\mathrm{\dots}+{b}_{s}{x}^{s})({c}_{0}+{c}_{1}x+\mathrm{\dots}+{c}_{t}{x}^{t})$$ 
with $s>0$ and $t>0$. Because the principal divisor $({a}_{0})$, i.e. $({b}_{0})({c}_{0})$ is divisible by the prime divisor $\U0001d52d$ and there is a unique factorisation in the monoid $\U0001d507$, $\U0001d52d$ must divide $({b}_{0})$ or $({c}_{0})$ but, by ${\U0001d52d}^{2}\nmid ({a}_{0})$, not both of $({b}_{0})$ and $({c}_{0})$; suppose e.g. that $\U0001d52d\mid {c}_{0}$. If $\U0001d52d$ would divide all the coefficients ${c}_{j}$, then it would divide also the product^{} ${b}_{s}{c}_{t}={a}_{n}$. So, there is a certain smallest index $k$ such that $p\nmid {c}_{k}$. Accordingly, in the sum ${b}_{0}{c}_{k}+{b}_{1}{c}_{k1}+\mathrm{\dots}+{b}_{k}{c}_{0}$, the prime divisor $\U0001d52d$ 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 ${a}_{k}$ which by hypothesis^{} is divisible by the prime divisor. This contradiction^{} shows that the polynomial $f(x)$ is irreducible.
Title  Eisenstein criterion in terms of divisor theory 

Canonical name  EisensteinCriterionInTermsOfDivisorTheory 
Date of creation  20130322 18:00:45 
Last modified on  20130322 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 