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
+ bx + c =
x) + c =
) + c -
= a(x +
We set (5) equal to 0 and we can easily solve for x. After dividing by a, we obtain
(6) (x +
The square roots of
are of course ±2a. Assuming that
- 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 +
b2 - 4ac
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
- 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.
Exercise 1.3. Show that additive and multiplicative inverses in a ﬁeld are
Exercise 1.4. A subtlety. Given a ﬁeld, is the formula
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
+x = 0,
+1 = 0, and
+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.