Linear Algebra, Theory And Applications, 2012a

210 VECTOR SPACES AND FIELDS
Corollary 8.3.11 Let q (λ) = ∏ p
i=1 φ i (λ) ki where the k i are positive integers and the φ i (λ)
are irreducible monic polynomials. Suppose also that p (λ) is a monic polynomial which
divides q (λ) . Then
p∏
p (λ) = φ i (λ) r i
i=1
where r i is a nonnegative integer no larger than k i .
Proof: Using Theorem 8.3.10, let p (λ) =b ∏ s
i=1 ψ i (λ) r i
where the ψ i (λ) areeach
irreducible and monic and b ∈ F. Sincep (λ) is monic, b =1. Then there exists a polynomial
g (λ) such that
s∏
p∏
p (λ) g (λ) =g (λ) ψ i (λ) r i
= φ i (λ) k i
Hence g (λ) must be monic. Therefore,
p (λ) g (λ) =
i=1
{
p(λ)
}} {
s∏
ψ i (λ) ri
i=1
i=1
l∏
η j (λ) =
j=1
p∏
φ i (λ) ki
for η j monic and irreducible. By uniqueness, each ψ i equals one of the φ j (λ) and the same
holding true of the η i (λ). Therefore, p (λ) is of the desired form.
8.3.2 Polynomials And Fields
When you have a polynomial like x 2 − 3 which has no rational roots, it turns out you can
enlarge the field of rational numbers to obtain a larger field such that this polynomial does
have roots in this larger field. I am going to discuss a systematic way to do this. It will
turn out that for any polynomial with coefficients in any field, there always exists a possibly
larger field such that the polynomial has roots in this larger field. This book has mainly
featured the field of real or complex numbers but this procedure will show how to obtain
many other fields which could be used in most of what was presented earlier in the book.
Here is an important idea concerning equivalence relations which I hope is familiar.
Definition 8.3.12 Let S be a set. The symbol, ∼ is called an equivalence relation on S if
it satisfies the following axioms.
1. x ∼ x for all x ∈ S. (Reflexive)
2. If x ∼ y then y ∼ x. (Symmetric)
3. If x ∼ y and y ∼ z, thenx ∼ z. (Transitive)
Definition 8.3.13 [x] denotes the set of all elements of S which are equivalent to x and
[x] is called the equivalence class determined by x or just the equivalence class of x.
Also recall the notion of equivalence classes.
Theorem 8.3.14 Let ∼ be an equivalence class defined on a set, S and let H denote the
set of equivalence classes. Then if [x] and [y] are two of these equivalence classes, either
x ∼ y and [x] =[y] or it is not true that x ∼ y and [x] ∩ [y] =∅.
i=1