Math 414: Analysis I
Fields
Definition 1 (Ordered Sets). An ordered set is a set A, together with a relation < such
that
1. For any x, y ∈ A, exactly one of x < y, x = y, y < x holds.
2. If x < y and y < z, then x < z.
Definition 2 (Field). A set F is called a field if it has two operations defined on it,
addition x + y, and multiplication xy, and it satisfies the following axioms:
1. (closure of addition) If x ∈ F and y ∈ F , then x + y ∈ F .
2. (commutativity of addition) If x + y = y + x for all x, y ∈ F .
3. (associativity of addition) If (x + y) + z = x + (y + z) for all x, y, z ∈ F .
4. There exists an element 0 ∈ F such that 0 + x = x for all x ∈ F .
5. For every element x ∈ F there exists an element −x ∈ F such that x + (−x) = 0.
6. (closure of multiplication) If x ∈ F , and y ∈ F , then xy ∈ F .
7. (commutativity of multiplication) If xy = yx for all x, y ∈ F .
8. (associativity of multiplication) If (xy)z = x(yz) for all x, y, z ∈ F .
9. There exists an element 1 (and 1 6= 0) such that 1x = x for all x ∈ F .
10. For every x ∈ F such that x 6= 0 there exists an element 1/x ∈ F such that
x(1/x) = 1.
11. (distributive law) x(y + z) = xy + xz for all x, y, z ∈ F .
Definition 3 (Ordered Field). A field F is said to be an ordered field if F is also an
ordered set such that:
1. For x, y, z ∈ F , x < y implies x + z < y + z.
2. For x, y ∈ F , x > 0 and y > 0 implies xy > 0.
We note that R, the set of all real numbers, is an ordered field. Thus, we may assume
that all of the properties mentioned above hold for real numbers.
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