3.4. POLYNOMIALS AND FIELDS 63

(a) Let a basis for ker(T ) be {z1, · · · ,zr} . Let a basis for Im(T ) be

{T v1, · · · ,T vs} .

You need to show that r + s = n. Begin with u ∈ V and consider Tu. It is alinear combination of {T v1, · · · ,T vs} say ∑

si=1 aiT vi. Why?

(b) Next explain why T (u−∑si=1 aivi) = 0. Then explain why there are scalars b j

such that u−∑si=1 aivi = ∑

rj=1 b jz j.

(c) Observe that V = span(z1, · · · ,zr,v1, · · · ,vs) . Why?

(d) Finally show that {z1, · · · ,zr,v1, · · · ,vs} is linearly independent. Thus n= r+s.

3.4 Polynomials and FieldsAs an application of the theory of vector spaces, this section considers the problem of fieldextensions. When you have a polynomial like x2− 3 which has no rational roots, it turnsout you can enlarge the field of rational numbers to obtain a larger field such that thispolynomial does have roots in this larger field. I am going to discuss a systematic way todo this. It will turn out that for any polynomial with coefficients in any field, there alwaysexists a possibly larger field such that the polynomial has roots in this larger field. Thisbook mainly features the field of real or complex numbers but this procedure will showhow to obtain many other fields. The ideas used in this development are the same as thoseused later in the material on linear transformations but slightly easier.

Here is an important idea concerning equivalence relations which I hope is familiar. Ifnot, see Section 1.3 on page 6.

Definition 3.4.1 Let S be a set. The symbol, ∼ is called an equivalence relation on S if itsatisfies 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, then x∼ z. (Transitive)

Definition 3.4.2 [x] denotes the set of all elements of S which are equivalent to x and [x] iscalled the equivalence class determined by x or just the equivalence class of x.

Also recall the notion of equivalence classes.

Theorem 3.4.3 Let∼ be an equivalence class defined on a set, S and let H denote the setof equivalence classes. Then if [x] and [y] are two of these equivalence classes, either x∼ yand [x] = [y] or it is not true that x∼ y and [x]∩ [y] = /0.

Definition 3.4.4 Let F be a field, for example the rational numbers, and denote by F [x] thepolynomials having coefficients in F. Suppose p(x) is a polynomial. Let a(x)∼ b(x) (a(x)is similar to b(x)) when

a(x)−b(x) = k (x) p(x)

for some polynomial k (x) . Denote by (p(x)) all polynomials of the form p(x)k (x) wherek (x) is some polynomial.