MATH 57091 - Algebra for High School Teachers
Unique Factorization of Polynomials
Professor Donald L. White
Department of Mathematical Sciences
Kent State University
D.L. White (Kent State University)
1 / 11
Review
In many ways, the irreducible polynomials in a polynomial ring F [x]
are analogous to the prime numbers in the ring of integers Z.
Our results on
unique factorization of integers, divisibility in terms of prime factorization,
and the description of the GCD and LCM in terms of prime factorization
all translate to statements about F [x] with only minor changes.
It may be helpful to review the following:
Fundamental Concepts of Algebra notes, §2.5 and §2.6
Week 3 - Episode 5: Fundamental Theorem of Arithmetic
Week 3 - Episode 6: GCD, LCM, and Prime Factorization
For details on the proofs of our results on polynomials,
see Fundamental Concepts of Algebra notes, §3.7.
D.L. White (Kent State University)
2 / 11
Review
Our unique factorization theorem for integers is
the Fundamental Theorem of Arithmetic:
Fundamental Theorem of Arithmetic
Every integer n > 1 is either prime or a product of primes.
The expression of n as a product of primes is unique
except for the order of the factors.
In particular, the theorem says that n > 1 can be written uniquely
in the form n = p1a1 p2a2 · · · prar , where p1 < p2 < · · · < pr are primes
and ai > 1 for all i (Canonical Prime Factorization of n).
D.L. White (Kent State University)
3 / 11
Review
We described divisibility of integers in terms of their prime factorizations:
Theorem 1
If a = p1a1 p2a2 · · · prar and b = p1b1 p2b2 · · · prbr , with the pi distinct primes
and ai > 0, bi > 0 for all i, then a | b if and only if ai 6 bi for all i.
Using this theorem, we are also able to describe the GCD and LCM
of two integers in terms of their prime factorizations:
Theorem 2
Let a = p1a1 p2a2 · · · prar and b = p1b1 p2b2 · · · prbr ,
with the pi distinct primes and ai > 0, bi > 0 for all i. Then
a
(a, b) = p1d1 p2d2 · · · prdr , where di = min{ai , bi } for all i, and
b
[a, b] = p1m1 p2m2 · · · prmr , where mi = max{ai , bi } for all i.
D.L. White (Kent State University)
4 / 11
Preliminary Results
Our first preliminary result is analogous to the fact that
an integer greater than 1 is either prime or has a prime factor.
Lemma 1
If F is a field and p(x) is a non-constant polynomial in F [x],
then either p(x) is irreducible or p(x) has an irreducible factor.
Sketch of Proof:
Proceed by induction on the degree of p(x).
We have seen that if deg p(x) = 1, then p(x) is irreducible.
If deg p(x) > 1 and p(x) is not irreducible, then p(x) = f (x)g (x),
where both f (x) and g (x) in F [x] are of lower degree than p(x).
By the inductive hypothesis, each of f (x) and g (x) is either irreducible
or has an irreducible factor, hence p(x) has an irreducible factor.
D.L. White (Kent State University)
5 / 11
Preliminary Results
Our next result is the polynomial version of Euclid’s Lemma for Primes
(if p is prime and p | ab, then p | a or p | b).
Lemma 2
Let F be a field and let p(x) be an irreducible polynomial in F [x].
If p(x) | f (x)g (x) with f (x), g (x) ∈ F [x], then p(x) | f (x) or p(x) | g (x).
Sketch of Proof:
If p(x) - f (x), then (p(x), f (x)) = 1 because p(x) is irreducible.
Hence
1 = a(x)p(x) + b(x)f (x)
for some a(x), b(x) ∈ F [x].
Multiply both sides of the equation by g (x) to obtain
g (x) = a(x)p(x)g (x) + b(x)f (x)g (x).
Now p(x) | p(x) and p(x) | f (x)g (x), and so p(x) | g (x)
by the Combination Theorem.
D.L. White (Kent State University)
6 / 11
Unique Factorization Theorem
The following theorem is the polynomial version of
the Fundamental Theorem of Arithmetic.
Unique Factorization Theorem
Let F be a field.
If f (x) is a non-constant polynomial in F [x], then f (x) is irreducible
or can be written as a product of irreducible polynomials in F [x].
Moreover, if
f (x) = p1 (x)p2 (x) · · · pk (x) = q1 (x)q2 (x) · · · qm (x),
where the pi (x), qj (x) are irreducible polynomials in F [x],
then k = m and, after possible rearrangement, pi (x) = ci qi (x)
for some non-zero ci ∈ F for all i.
Sketch of Proof:
The first statement is proved using Lemma 1 and induction on deg f (x).
The second statement is proved using Lemma 2.
See FCA notes, Theorem 3.7.5, for a complete proof.
D.L. White (Kent State University)
7 / 11
Unique Factorization Theorem
NOTES:
The Unique Factorization Theorem
is true in R[x] for some more general rings R.
What is necessary is that the unique factorization property hold in R.
In particular, the theorem is also true in Z[x].
The theorem does not hold in Z12 , for example:
x 2 − 5x + 6 = (x − 2)(x − 3)
= (x + 1)(x + 6)
because −5 = 7 in Z12 .
The theorem implies that if f (x) is not irreducible, then
f (x) = α · p1 (x)a1 p2 (x)a2 · · · pr (x)ar ,
where α is the leading coefficient of f (x), the pi (x) are distinct,
irreducible, monic polynomials in F [x], and ai > 1 for all i.
D.L. White (Kent State University)
8 / 11
Divisibility, GCD, and LCM
The next result is the polynomial version of Theorem 1 on divisibility.
Theorem
Let F be a field and let
a(x) = αp1 (x)a1 p2 (x)a2 · · · pr (x)ar ,
b(x) = βp1 (x)b1 p2 (x)b2 · · · pr (x)br ,
with the pi (x) distinct, monic, irreducible polynomials in F [x],
α and β non-zero elements of F , and ai > 0, bi > 0 for all i.
Then a(x) | b(x) if and only if ai 6 bi for all i.
D.L. White (Kent State University)
9 / 11
Divisibility, GCD, and LCM
We can define the LCM of polynomials, using degree as a measure of size.
Definition
Let F be a field and let a(x) and b(x) be non-zero polynomials in F [x].
The Least Common Multiple of a(x) and b(x)
is the monic polynomial m(x) ∈ F [x], denoted m(x) = [a(x), b(x)],
satisfying
i
a(x) | m(x) and b(x) | m(x), and
ii
if a(x) | c(x) and b(x) | c(x) for a non-zero polynomial c(x) in F [x],
then deg m(x) 6 deg c(x).
D.L. White (Kent State University)
10 / 11
Divisibility, GCD, and LCM
Finally, we have the polynomial version of Theorem 2 on the GCD & LCM.
Theorem
Let F be a field and let
a(x) = αp1 (x)a1 p2 (x)a2 · · · pr (x)ar ,
b(x) = βp1 (x)b1 p2 (x)b2 · · · pr (x)br ,
with the pi (x) distinct, monic, irreducible polynomials in F [x],
α and β non-zero elements of F , and ai > 0, bi > 0 for all i. Then
i
ii
(a(x), b(x)) = p1 (x)d1 p2 (x)d2 · · · pr (x)dr ,
where di = min{ai , bi } for all i, and
[a(x), b(x)] = p1 (x)m1 p2 (x)m2 · · · pr (x)mr ,
where mi = max{ai , bi } for all i.
D.L. White (Kent State University)
11 / 11