6 1. From the Real Numbers to the Complex Numbers

Proof. The idea of the proof is to complete the square. Since both a and 2 are

nonzero elements of F, they have multiplicative inverses. We therefore have

ax2

+ bx + c =

a(x2

+

b

a

x) + c =

a(x2

+

b

a

x +

b2

4a2

) + c -

b2

4a

= a(x +

b

2a

)2

+

4ac -

b2

4a

. (5)

We set (5) equal to 0 and we can easily solve for x. After dividing by a, we obtain

(6) (x +

b

2a

)2

=

b2

- 4ac

4a2

.

The square roots of

4a2

are of course ±2a. Assuming that

b2

- 4ac has a square

root in F, we solve (6) for x by ﬁrst taking the square root of both sides. We obtain

(7) x +

b

2a

= ±

√

b2 - 4ac

2a

.

After a subtraction and simpliﬁcation we obtain (4) from (7).

The reader surely has seen the quadratic formula before. Given a quadratic

polynomial with real coeﬃcients, the formula tells us that the polynomial will have

no real roots when

b2

- 4ac 0. For many readers the ﬁrst exposure to complex

numbers arises when we introduce square roots of negative numbers in order to use

the quadratic formula.

I

Exercise 1.3. Show that additive and multiplicative inverses in a ﬁeld are

unique.

I

Exercise 1.4. A subtlety. Given a ﬁeld, is the formula

√

u

√

v =

√

uv

always valid? In the proof of the quadratic formula, did we use this formula im-

plicitly? If not, what did we use?

Example 2.2. One can completely analyze quadratic equations with coeﬃcients in

F2. The only such equations are

x2

= 0,

x2

+x = 0,

x2

+1 = 0, and

x2

+x+1 = 0.

The ﬁrst equation has only the solution 0. The second has the two solutions 0 and

1. The third has only the solution 1. The fourth has no solutions. We have given

a complete analysis, even though Theorem 2.1 cannot be used in this setting.

Before introducing the notion of ordered ﬁeld, we give a few other examples of

ﬁelds. Several of these examples use modular (clock) arithmetic. The phrases add

modulo p and multiply modulo p have the following meaning. Fix a positive integer

p, called the modulus. Given integers m and n, we add (or multiply) them as usual

and then take the remainder upon division by p. The remainder is called the sum

(or product) modulo p. This natural notion is familiar to everyone; ﬁve hours after

nine o’clock is two o’clock; we added modulo twelve. The subsequent examples can

be skipped without loss of continuity.

Example 2.3. Fields with ﬁnitely many elements. Let p be a prime number,

and let Fp consist of the numbers 0, 1,...,p - 1. We deﬁne addition and multiplica-

tion modulo p. Then Fp is a ﬁeld.