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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.)
 (Associativity of addition.) Addition (+) is an
associative operation on F.
 (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.
 (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.
 (Commutativity of multiplication.) Multiplication (^{.}) is a
commutative operation on F.
 (Associativity of multiplication.) Multiplication is an associative
operation on F.
 (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.
 (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.
 (Distributive law.) For all x, y, z in F, x^{.}(y+z)=
(x^{.}y)+(x^{.}z).
 (Zeroone 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
 (Sum of positives is positive.) For all a,b
in F^{+}, a+b is in
F^{+}
 (Product of positives is positive.) For all a,b
in F^{+}, a^{.}b
is in F^{+}.
 (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:

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