1.4. ORDERED FIELDS 11

Axiom 1.3.5 xy = yx, (commutative law for multiplication).

Axiom 1.3.6 (xy) z = x (yz) , (associative law for multiplication).

Axiom 1.3.7 1x = x, (multiplicative identity).

Axiom 1.3.8 For each x ̸= 0, there exists x−1 such that xx−1 = 1.(existence of multiplica-tive inverse).

Axiom 1.3.9 x (y + z) = xy + xz.(distributive law).

These axioms are known as the field axioms and any set (there are many others besidesR) which has two such operations satisfying the above axioms is called a field. Division andsubtraction are defined in the usual way by x−y ≡ x+(−y) and x/y ≡ x

(y−1

).We assume

0 ̸= 1 so that the axioms will describe someting useful.Here is a little proposition which derives some familiar facts.

Proposition 1.3.10 0 and 1 are unique. Also −x is unique and x−1 is unique. Further-more, 0x = x0 = 0 and −x = (−1)x.

Proof: Suppose 0′ is another additive identity. Then

0′ = 0′ + 0 = 0.

Thus 0 is unique. Say 1′ is another multiplicative identity. Then

1 = 1′1 = 1′.

Now suppose y acts like the additive inverse of x. Then

−x = (−x) + 0 = (−x) + (x+ y) = (−x+ x) + y = y

Finally,0x = (0 + 0)x = 0x+ 0x

and so0 = − (0x) + 0x = − (0x) + (0x+ 0x) = (− (0x) + 0x) + 0x = 0x

Finallyx+ (−1)x = (1 + (−1))x = 0x = 0

and so by uniqueness of the additive inverse, (−1)x = −x. ■

1.4 Ordered Fields

The real numbers R are an example of an ordered field. More generally, here is a definition.

Definition 1.4.1 Let F be a field. It is an ordered field if there exists an order, < whichsatisfies

1. For any x, y, exactly one of the following holds: x = y, x < y, or y < x.

2. If x < y and either z < w or z = w, then, x+ z < y + w.

3. If 0 < x, 0 < y, then xy > 0.