This result is very useful for producing examples of
irreducible
polynomials.
Theorem
Let f=anxn + an-1xn-1 + ... + a0
be a nonconstant polynomial with integer coefficients and let
p be a prime number.
Suppose that
- p does not divide an
-
p|an-1,...,a0
-
p2 doesn't
divide a0.
Then
f is an irreducible polynomial
in
Q[x].
Examples
-
x6 - 30x5 + 6x4 - 18x3 +
12x2 - 6x +12
is irreducible in Q[x] by Eisenstein
with p=3 (note that we can't use p=2 :)
-
xn - 2 is irreducible by Eisenstein with p=2.
-
Consider f(x)=x3- 3x - 1. We can't apply Eisenstein
directly but consider f(x+1) (Obviously if f(x+1) is irreducible
then so is f(x).) We have
f(x+1)=(x+1)3 - 3(x+1) - 1 = x3 + 3x2 -3.
By Eisenstein (p=3) we deduce that f(x) is irreducible.
In fact Eisenstein's criterion is a special case of a more general result.
Theorem
Let R be a unique factorization domain with field of fractions
K. Let f=anxn + an-1xn-1 + ... + a0
be a nonconstant polynomial in R[x]. Let p be a
prime in R.
Suppose that
- p does not divide an
-
p|an-1,...,a0
-
p2 doesn't
divide a0.
Then
f is an irreducible polynomial
in
K[x].
Proof of Eisenstein's criterion:
Firstly, we can assume that f is primitive, for if we write
f=ch, with c the content and h primitive then
since p doesn't divide an it doesn't divide
c. It follows quickly that h also satisfies the conditions
of the criterion. Finally, if h is irreducible then so is
f.
By Gauss's Lemma, if f fails to be irreducible in
K[x] then it has a factorization
f=f1f2 in
R[x] so that f1,f2
both have degree < deg f.
Let's say that f1=c0+ ... crxr
and
f2=d0+ ... drxr.
Now a0=c0d0 and a0
is divisible by p but not by p2. Thus one of
c0 and d0 is divisble by p
and the other is not. WLOG p|c0.
but doesn't divide d0.
Now p does not divide an=crds
so it doesn't divide cr. Let k be the smallest
integer such that p does not divide ck.
Thus, k>0 and k<=r. Now
ak=c0dk+ ... + ckd0.
We know that p|ak and p|c0,...,ck-1
so it follows that p|ckd0. But p
doesn't divide either of the two terms in this product and so this contradicts
the primeness of p, completing the proof.