20 1. From the Real Numbers to the Complex Numbers

Given an equivalence relation

∼

=

on S, we partition S into equivalence classes.

All the elements in a single equivalence class are equivalent, and no other member

of S is equivalent to any of these elements. We have already seen two elementary

examples. First, fractions

a

b

and

c

d

are equivalent if and only if they represent

the same real number, that is, if and only if ad = bc. Thus a rational number

may be regarded as an equivalence class of pairs of integers. Second, when doing

arithmetic modulo p, we regard two integers as being in the same equivalence class

if their diﬀerence is divisible by p.

I

Exercise 1.29. The precise deﬁnition of modular arithmetic involves equivalence

classes; we add and multiply equivalence classes (rather than numbers). Show that

addition and multiplication modulo p are well-deﬁned concepts. In other words, do

the following. Assume m1 and m2 are in the same equivalence class modulo p and

that n1 and n2 are also in the same equivalence class (not necessarily the same class

m1 and m2 are in). Show that m1 + n1 and m2 + n2 are in the same equivalence

class modulo p. Do the same for multiplication.

I

Exercise 1.30. Let S be the set of students at a college. For s,t ∈ S, consider

the relation s

∼

=

t if s and t take a class together. Is this relation an equivalence

relation?

Let R[t] denote the collection of polynomials in one variable, with real coeﬃ-

cients. An element p of R[t] can be written

p =

Xd

j=0

ajtj,

where aj ∈ R. Notice that the sum is ﬁnite. Unless all the aj are 0, there is a largest

d for which aj 6 = 0. This number d is called the degree of the polynomial. When all

the aj equal 0, we call the resulting polynomial the zero polynomial and agree that

it has no degree. (In some contexts, one assigns the symbol -∞ to be the degree

of the zero polynomial.) The sum and the product of polynomials are deﬁned as in

high school mathematics. In many ways R[t] resembles the integers Z. Each is a

commutative ring under the operations of sum and product. Unique factorization

into irreducible elements holds in both settings, and the division algorithm works

the same as well. See [4] or [8] for more details. Given polynomials p and g, we say

that p is a multiple of g, or equivalently that g divides p, if there is a polynomial q

with p = gq.

The polynomial 1+

t2

is irreducible, in the sense that it cannot be written as a

product of two polynomials, each of lower degree, with real coeﬃcients. The set I

of polynomials divisible by 1 +

t2

is called the ideal generated by 1 +

t2.

Given two

polynomials p,q, we say that they are equivalent modulo I if p - q ∈ I, in other

words, if p - q is divisible by 1 +

t2.

We observe that the three properties of an

equivalence relation hold:

• For all p, p

∼

= p.

• For all p,q, p

∼

=

q if and only if q

∼

=

p.

• If p

∼

=

q and q

∼

=

r, then p

∼

=

r.