Irreducible Polynomial
Let D be an integral domain.
A polynomial f ( x) from D[ x] that is neither the zero
polynomial nor a unit in D[ x] , such that whenever f ( x)
is expressed as a product f ( x)  g ( x)h( x) , with g ( x)
and h( x) from D[ x] , then g ( x) or h( x) is a unit in
D[ x] .
Reducible Polynomial
Let D be an integral domain.
A nonzero, nonunit polynomial from D[ x] that is not
irreducible.
Reducibility Test for Degrees 2 and 3
Let F be a field. If f ( x)  F[ x] and deg f ( x)  2 or 3,
then f ( x) is reducible over F if and only if f ( x) has a
zero in F.
Content of a polynomial
Polynomial coefficients are integer.
The greatest common divisor of the polynomial
coefficients.
Primitive Polynomial
A polynomial in Z [ x] with content 1.
Gauss’s Lemma
The product of two primitive polynomials is primitive.
Over Q Implies Over Z
Let f ( x)  Z[ x] . If f ( x) is reducible over Q, then it is
reducible over Z.
Mod p Irreducibility Test
Let p be a prime and suppose that f ( x)  Z[ x] with
deg f ( x) 1. Let f ( x) be the polynomial in Z p[ x]
obtained from f ( x) by reducing all of its coefficients
mod p . If f ( x) is irreducible over Z p and
deg f ( x)  deg f ( x) , then f ( x) is irreducible over Q.
Eisenstein’s Criterion
Let f ( x)  an xn  ...  a0  Z [ x] . If there is a prime p
such that p divides every coefficient of f ( x) except for
an and p 2 does not divide a0 , then f ( x) is irreducible
over Q.
Irreducibility of p-th Cyclotomic Polynomial
For any prime p, the p-th cyclotomic polynomial
p
x
 p ( x)  1  x p1  x p2  ...  x 1
x 1
is irreducible over Q.
p( x) Is Irreducible if and Only if  p( x)  is Maximal
Let F be a field and p( x)  F[ x] . Then  p( x)  is a
maximal ideal in F [ x] if and only if p( x) is irreducible
over F.
F[ x]/  p( x)  Is a Field
Let F be a field and p( x) an irreducible polynomial over
F. Then F[ x]/  p( x)  is a field.
p( x)| a( x)b( x) Implies p( x)| a( x) or p( x)| b( x)
Let F be a field and let p( x), a( x), b( x)  F[ x] . If p( x)
is irreducible over F and p( x)| a( x)b( x) , then p( x)| a( x)
or p( x)| b( x) .
Unique Factorization in Z [ x]
Every polynomial in Z [ x] that is not the zero polynomial
or a unit in Z [ x] can be written in the form
b1b2...bs p1( x) p2 ( x)... pm ( x) , where the bi ’s are
irreducible polynomials of degree 0, and the pi ( x) ’s are
irreducible polynomials of positive degree.
Furthermore, if b1b2...bs p1( x) p2 ( x)... pm ( x) 
c1c2...ct q1( x)q2 ( x)...qn ( x) , where the ci ’s are irreducible
polynomials of degree 0, and the qi ( x) ’s are irreducible
polynomials of positive degree, then s  t , m  n , and,
after renumbering the c’s and q(x)’s, we have bi  ci
for i  1,..., s and pi ( x)  qi ( x) for i  1,..., m .