3. Inequalities and ordered ﬁelds 7

In Example 2.3, p needs to be a prime number. Property 5) of Proposition 2.1

fails when p is not a prime. We mention without proof that the number of elements

in a ﬁnite ﬁeld must be a power of a prime number. Furthermore, for each prime

p and positive integer n, there exists a ﬁnite ﬁeld with

pn

elements.

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Exercise 1.5. True or false? Every quadratic equation in F3 has a solution.

Fields such as Fp are important in various parts of mathematics and computer

science. For us, they will serve only as examples of ﬁelds. The most important

examples of ﬁelds for us will be the real numbers and the complex numbers. To

deﬁne these ﬁelds rigorously will take a bit more eﬀort. We end this section by

giving an example of a ﬁeld built from the real numbers. We will not use this

example in the logical development.

Example 2.4. Let K denote the collection of rational functions in one variable

x with real coeﬃcients. An element of K can be written

p(x)

q(x)

, where p and q are

polynomials, and we assume that q is not the zero polynomial. (We allow q(x) to

equal 0 for some x, but not for all x.) We add and multiply such rational functions

in the usual way. It is tedious but not diﬃcult to verify the ﬁeld axioms. Hence

K is a ﬁeld. Furthermore, K contains R in a natural way; we identify the real

number c with the constant rational function

c

1

. As with the rational numbers,

many diﬀerent fractions represent the same element of K. To deal rigorously with

such situations, one needs the notion of equivalence relation, discussed in Section 5.

3. Inequalities and ordered ﬁelds

Comparing the sizes of a pair of integers or of a pair of rational numbers is both

natural and useful. It does not make sense however to compare the sizes of elements

in an arbitrary ring or ﬁeld. We therefore introduce a crucial property shared by

the integers Z and the rational numbers Q. For x,y in either of these sets, it makes

sense to say that x y. Furthermore, given the pair x,y, one and only one of three

things must be true: x y, x y, or x = y. We need to formalize this idea in

order to deﬁne the real numbers.

Deﬁnition 3.1. A ﬁeld F is called ordered if there is a subset P ⊂ F, called the

set of positive elements of F, satisfying the following properties:

1) For all x,y in P, we have x + y ∈ P and xy ∈ P (closure).

2) For each x in F, one and only one of the following three statements is true:

x = 0, x ∈ P, -x ∈ P (trichotomy).

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Exercise 1.6. Let F be an ordered ﬁeld. Show that 1 ∈ P.

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Exercise 1.7. Show that the trichotomy property can be rewritten as follows.

For each x,y in F, one and only one of the following three statements is true: x = y,

x - y ∈ P, y - x ∈ P.

The rational number system is an ordered ﬁeld; a fraction

p

q

is positive if and

only if p and q have the same sign. Note that q is never 0 and that a rational

number is 0 whenever its numerator is 0. It is of course elementary to check in