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
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.
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.
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
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 =
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+
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 +
is called the ideal generated by 1 +
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 +
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

If p

q and q

r, then p

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