64 CHAPTER 3. VECTOR SPACES

Proposition 3.4.5 In the above definition, ∼ is an equivalence relation.

Proof: First of all, note that a(x) ∼ a(x) because their difference equals 0p(x) . Ifa(x)∼ b(x) , then a(x)−b(x) = k (x) p(x) for some k (x) . But then

b(x)−a(x) =−k (x) p(x)

and so b(x) ∼ a(x). Next suppose a(x) ∼ b(x) and b(x) ∼ c(x) . Then a(x)− b(x) =k (x) p(x) for some polynomial k (x) and also b(x)−c(x) = l (x) p(x) for some polynomiall (x) . Then

a(x)− c(x) = a(x)−b(x)+b(x)− c(x)

= k (x) p(x)+ l (x) p(x) = (l (x)+ k (x)) p(x)

and so a(x)∼ c(x) and this shows the transitive law. ■

Definition 3.4.6 Let F be a field and let p(x) ∈ F [x] be a nonzero monic polynomial. Thismeans that the coefficient of the highest power is 1. Also let p(x) have degree at least1. For the similarity relation of Definition 3.4.4, define the following operations on theequivalence classes. [a(x)] is an equivalence class means that it is the set of all polynomialswhich are similar to a(x).

[a(x)]+ [b(x)]≡ [a(x)+b(x)]

[a(x)] [b(x)]≡ [a(x)b(x)]

This collection of equivalence classes is sometimes denoted by F [x]/(p(x)). This is calleda quotient space.

The set of equivalence classes just described is a commutative ring. This is like afield except it may fail to have multiplicative inverses. The reason for considering onlypolynomials of degree at least 1 is that F [x]/(1) isn’t very interesting because f (x)∼ g(x)if and only if their difference is a multiple of 1. Thus every two polynomials are similar sothere is only one similarity class. In particular, [1]∼ [0] . It is shown below that this is welldefined.

Axiom 3.4.7 Here are the axioms for a commutative ring.

1. x+ y = y+ x, (commutative law for addition)

2. There exists 0 such that x+0 = x for all x, (additive identity).

3. For each x ∈ F, there exists −x ∈ F such that x+(−x) = 0, (existence of additiveinverse).

4. (x+ y)+ z = x+(y+ z) ,(associative law for addition).

5. xy = yx,(commutative law for multiplication). You could write this as x× y = y× x.

6. (xy)z = x(yz) ,(associative law for multiplication).

7. There exists 1 such that 1x = x for all x,(multiplicative identity).

8. x(y+ z) = xy+ xz.(distributive law).