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# 2.3 The Field Axioms

2.48   Definition (Field.) A field is a triple where is a set, and and are binary operations on (called addition and multiplication respectively) satisfying the following nine conditions. (These conditions are called the field axioms.)
2. (Existence of additive identity.) There is an identity element for addition.

We know that this identity is unique, and we will denote it by .

3. (Existence of additive inverses.) Every element of is invertible for .

We know that the additive inverse for is unique, and we will denote it by .

4. (Commutativity of multiplication.) Multiplication is a commutative operation on .
5. (Associativity of multiplication.) Multiplication is an associative operation on .
6. (Existence of multiplicative identity.) There is an identity element for multiplication.

We know that this identity is unique, and we will denote it by .

7. (Existence of multiplicative inverses.) Every element of except possibly for is invertible for .

We know that the multiplicative inverse for is unique, and we will denote it by . We do not assume is not invertible. We just do not assume that it is.

8. (Distributive law.) For all in , .
9. (Zero-one law.) The additive identity and multiplicative identity are distinct; i.e., .

We often speak of  the field " instead of  the field ".

2.49   Remark. Most calculus books that begin with the axioms for a field (e.g., [47, p5], [1, p18], [13, p5], [12, p554]) add an additional axiom.
10.
I have omitted this because, as Leonard Dickson pointed out in 1905[18, p202], it can be proved from the other axioms (see theorem 2.72 for a proof). I agree with Aristotle that
It is manifest that it is far better to make the principles finite in number. Nay, they should be the fewest possible provided they enable all the same results to be proved. This is what mathematicians insist upon; for they take as principles things finite either in kind or in number[26, p178].

2.50   Remark (Parentheses.) The distributive law is usually written as
 (2.51)

The right side of (2.51) is ambiguous. There are five sensible ways to interpret it:

The conventions presently used for interpreting ambiguous statements such as and involving operations are:
1. Multiplication and division have equal precedence.
2. Addition and subtraction have equal precedence.
3. Multiplication has higher precedence than addition.
This means that to interpret
 (2.52)

you first read (2.52) from left to right and perform all the multipliations and divisions as you come to them, getting
 (2.53)

Then read (2.53) from left to right performing all additions and subtractions as you come to them, getting

When I was in high school, multiplication had higher precedence than division, so

meant

whereas today it means

In 1713, addition often had higher precedence than multiplication. Jacob Bernoulli [8, p180] wrote expressions like

to mean

2.54   Examples. with the usual operations of addition and multiplication is a field.

is a field. (See definition 2.42 for the definitions.) We showed in section 2.2 that satisfies all the field axioms except possibly the distributive law. In appendix B, it is shown that the distributive property holds for for all , . (The proof assumes that the distributive law holds in .)

For a general , , the only field axiom that can possibly fail to hold in is the existence of multiplicative inverses, so to determine whether is a field, it is just necessary to determine whether every non-zero element in is invertible for .

2.55   Exercise. A In each of the examples below, determine which field axioms are valid and which are not. Which examples are fields? In each case that an axiom fails to hold, give an example to show why it fails to hold.
1. a) where and are usual addition and multiplication.
2. b) where is the set of non-negative rational numbers, and and are the usual addition and multiplication.
3. c) where is a set with just one element and both and are the only binary operation on ; i.e.,

4. d) where both and are the usual operation of addition on , e.g., and .

2.56   Exercise. Determine for which values of , is a field. (You already know that produces a field.)

2.57   Notation (The field .) Let , be a number such that is a field. Then  the field " means the field . I will often denote the operations in by and instead of and .

2.58   Entertainment. Determine for which values of in the system is a field. If you do this you will probably conjecture the exact (fairly simple) condition on that makes the system into a field.

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