6. A glimpse at metric spaces 21

This equivalence relation partitions the set R[t] into equivalence classes; the

situation is strikingly similar to modular arithmetic. Given a polynomial p(t), we

use the division algorithm to write p(t) = q(t)(1 +

t2)

+ r(t), where the remainder

r has degree at most one. Thus r(t) = a + bt for some a,b, and this r is the

unique ﬁrst-degree polynomial equivalent to p. In the case of modular arithmetic

we used the remainder upon division by the modulus; here we use the remainder

upon division by

t2

+ 1.

I

Exercise 1.31. Verify the transitivity property of equivalence modulo I.

Standard notation in algebra writes R[t]/(1 +

t2)

for the set of equivalence

classes. We can add and multiply in R[t]/(1 +

t2).

As usual, the sum (or product)

of equivalence classes P and Q is deﬁned to be the equivalence class of the sum

p + q (or the product pq) of members; the result is independent of the choice. An

equivalence class then can be identiﬁed with a polynomial a + bt, and the sum and

product of equivalence classes satisﬁes (15) and (16). In this setting we deﬁne C

as the collection of equivalence classes with this natural sum and product:

(25) C = R[t]/(1 +

t2).

Deﬁnition (25) allows us to set t2 = -1 whenever we encounter a term of

degree at least two. The irreducibility of t2 + 1 matters. If we form R[t]/(p(t)) for

a reducible polynomial p, then the resulting object will not be a ﬁeld. The reason

is precisely parallel to the situation with modular arithmetic. If we consider Z/(n),

then we get a ﬁeld (written Fn) if and only if n is prime.

I

Exercise 1.32. Show that R[t]/(t3 + 1) is not a ﬁeld.

I

Exercise 1.33. A polynomial

∑d

k=0

cktk in R[t] is equivalent to precisely one

polynomial of the form A + Bt in the quotient space. What is A + Bt in terms of

the coeﬃcients ck?

I

Exercise 1.34. Prove the division algorithm in R[t]. In other words, given

polynomials p and g, with g not the zero polynomial, show that one can write

p = qg + r where either r = 0 or the degree of r is less than the degree of g. Show

that q and r are uniquely determined by p and g.

I

Exercise 1.35. For any polynomial p and any x0, show that there is a polyno-

mial q such that p(x) = (x - x0)q(x) + p(x0).

6. A glimpse at metric spaces

Both the real number system and the complex number system provide intuition for

the general notion of a metric space. This section can be omitted without impacting

the logical development, but it should appeal to some readers.

Deﬁnition 6.1. Let X be a set. A distance function on X is a function δ : X×X →

R such that the following hold:

1) δ(x,y) ≥ 0 for all x,y ∈ X (distances are nonnegative).

2) δ(x,y) = 0 if and only if x = y (distinct points have positive distance between

them; a point has 0 distance to itself).