Eisenstein's irreducibility criterion is a sufficient condition assuring that an integer polynomial 
is irreducible in the polynomial
ring ![Q[x]](/images/equations/EisensteinsIrreducibilityCriterion/Inline2.svg)
.
The polynomial
 |
where 
for all 
and 
(which means that the degree of 
is 
) is irreducible if some prime number 
divides all coefficients 
, ..., 
, but not the leading coefficient 
and, moreover, 
does not divide the constant term 
.
This is only a sufficient, and by no means a necessary condition. For example, the polynomial 
is irreducible, but does not fulfil the above property,
since no prime number divides 1. However, substituting 
for 
produces the polynomial 
, which does fulfill the Eisenstein criterion (with
)
and shows the polynomial is irreducible.
