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2.48 Definition (Field.) A field is a triple (F,+,.) where F is a set, and + and . are binary operations on F (called addition and multiplication respectively) satisfying the following nine conditions. (These conditions are called the field axioms.)
  1. (Associativity of addition.) Addition (+) is an associative operation on F.
  2. (Existence of additive identity.) There is an identity element for addition.
    We know from (2.7) that this identity is unique, and we will denote it by 0.
  3. (Existence of additive inverses.) Every element x of F is invertible for +.
    We know from (2.15) that the additive inverse for x is unique, and we will denote it by -x.
  4. (Commutativity of multiplication.) Multiplication (.) is a commutative operation on F.
  5. (Associativity of multiplication.) Multiplication is an associative operation on F.
  6. (Existence of multiplicative identity.) There is an identity element for multiplication.
    We know from (2.7) that this identity is unique, and we will denote it by 1.
  7. (Existence of multiplicative inverses.) Every element x of F except possibly for 0 is invertible for ..
    We know from (2.15) that the multiplicative inverse for x is unique, and we will denote it by x-1. We do not assume 0 is not invertible. We just do not assume that it is.
  8. (Distributive law.) For all x, y, z in F, x.(y+z)= (x.y)+(x.z).
  9. (Zero-one law.) The additive identity and multiplicative identity are distinct; i.e. 0 is not equal to 1.
We will show later that it follows from these assumptions that addition is commutative.

2.100 Definition (Ordered field axioms.) An ordered field is a pair (F,F+) =((F,+,.),F+) where F is a field, and F+ is a subset of F satisfying the conditions

  1. (Sum of positives is positive.) For all a,b in F+, a+b is in F+
  2. (Product of positives is positive.) For all a,b in F+, a.b is in F+.
  3. (Trichotomy law.) For all a in F, exactly one of the statements

    a is in F+, -a is in F+, a=0

    is true.
The set F+> is called the set of positive elements of F.

5.21 Definition (Completeness axiom.) Let F be an ordered field. We say that F is complete, if it satisfies the condition:

  1. Every binary search sequence in F converges to a unique point in F.

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