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, < which
- For any x≠y, either x < y or y < x.
- If x < y and either z < w or z = w, then, x + z < y + w.
- If 0 < x,0 < y, then xy > 0.
With this definition, the familiar properties of order can be proved. The following proposition
lists many of these familiar properties. The relation ‘a > b’ has the same meaning as
‘b < a’.
Proposition 1.4.2 The following are obtained.
- If x < y and y < z, then x < z.
- If x > 0 and y > 0, then x + y > 0.
- If x > 0, then −x < 0.
- If x≠0, either x or −x is > 0.
- If x < y, then −x > −y.
- If x≠0, then x2 > 0.
- If 0 < x < y then x−1 > y−1.
Proof: First consider 1, called the transitive law. Suppose that x < y and y < z. Then from
the axioms, x + y < y + z and so, adding −y to both sides, it follows
Next consider 2. Suppose x > 0 and y > 0. Then from 2,
Next consider 3. It is assumed x > 0 so
Now consider 4. If x < 0, then
Consider the 5. Since x < y, it follows from 2
and so by 4 and Proposition 1.3.10,
Also from Proposition 1.3.10
Consider 6. If x > 0, there is nothing to show. It follows from the definition. If x < 0, then by
4, −x > 0 and so by Proposition 1.3.10 and the definition of the order,
By this proposition again,
= 1 and so
0 as claimed. Note that 1 >
because it equals 12
Finally, consider 7. First, if x > 0 then if x−1 < 0, it would follow
0 and so
0. However, this would require
from what was just shown. Therefore, x−1 > 0. Now the assumption implies y +
0 and so
multiplying by x−1,
Now multiply by y−1, which by the above satisfies y−1 > 0, to obtain
In an ordered field the symbols ≤ and ≥ have the usual meanings. Thus a ≤ b means a < b or
else a = b, etc.