Linear Algebra, Theory And Applications, 2012a

8.3. LOTS OF FIELDS 209
Lemma 8.3.9 Suppose p (λ) is a monic polynomial and q (λ) is a polynomial such that
p (λ) q (λ) =0.
Then q (λ) =0. Also if
then q 1 (λ) =q 2 (λ) .
p (λ) q 1 (λ) =p (λ) q 2 (λ)
Proof: Let
p (λ) =
Then the product equals
k∑
n∑
p j λ j ,q(λ) = q i λ i ,p k =1.
j=1
j=1 i=1
i=1
k∑ n∑
p j q i λ i+j .
Then look at those terms involving λ k+n . This is p k q n λ k+n and is given to be 0.
p k =1, it follows q n =0. Thus
k∑
n−1
∑
p j q i λ i+j =0.
j=1 i=1
Since
Then consider the term involving λ n−1+k and conclude that since p k =1, it follows q n−1 =0.
Continuing this way, each q i =0. This proves the first part. The second follows from
p (λ)(q 1 (λ) − q 2 (λ)) = 0.
The following is the analog of the fundamental theorem of arithmetic for polynomials.
Theorem 8.3.10 Let f (λ) be a nonconstant polynomial with coefficients in F. Then there
is some a ∈ F such that f (λ) =a ∏ n
i=1 φ i (λ) where φ i (λ) is an irreducible nonconstant
monic polynomial and repeats are allowed. Furthermore, this factorization is unique in the
sense that any two of these factorizations have the same nonconstant factors in the product,
possibly in different order and the same constant a.
Proof: That such a factorization exists is obvious. If f (λ) is irreducible, you are done.
Factor out the leading coefficient. If not, then f (λ) =aφ 1 (λ) φ 2 (λ) where these are monic
polynomials. Continue doing this with the φ i and eventually arrive at a factorization of the
desired form.
It remains to argue the factorization is unique except for order of the factors. Suppose
n∏
m∏
a φ i (λ) =b ψ i (λ)
i=1
where the φ i (λ) andtheψ i (λ) are all irreducible monic nonconstant polynomials and a, b ∈
F. If n>m,then by Lemma 8.3.8, each ψ i (λ) equals one of the φ j (λ) . By the above
cancellation lemma, Lemma 8.3.9, you can cancel all these ψ i (λ) with appropriate φ j (λ)
and obtain a contradiction because the resulting polynomials on either side would have
different degrees. Similarly, it cannot happen that n