2.3 The Field Axioms

2.48   Definition (Field.) A field is a triple $(F,+,\cdot)$ where $F$ is a set, and $+$ and $\cdot$ 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 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 that the additive inverse for $x$ is unique, and we will denote it by $-x$.

4. (Commutativity of multiplication.) Multiplication $(\cdot)$ 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 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 $\cdot$.

   We know 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 $F$, $x\cdot(y+z)=(x\cdot y)+(x\cdot z)$.
9. (Zero-one law.) The additive identity and multiplicative identity are distinct; i.e., $0\neq 1$.

We often speak of `` the field $F$" instead of `` the field $(F,+,\cdot)$".

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.

(Commutativity of addition.) Addition is a commutative operation on $F$.

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

$$x\cdot (y+z)=x\cdot y+x\cdot z$$ (2.51)

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

$$\begin{eqnarray*} &\;&x\cdot\left((y+x)\cdot z\right), \\ &\;&x\cdot\left(y+(x\... ...dot y)+x\right)\cdot z \\ &\;&\left(x\cdot (y+x)\right)\cdot z. \end{eqnarray*}$$

The conventions presently used for interpreting ambiguous statements such as $x\cdot y+x\cdot z$ and involving operations $+,-,\cdot,/$ 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

$$1\cdot 2 / 3+4\cdot 5\cdot 6-7\cdot 8+9,$$ (2.52)

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

$$\left((1\cdot 2)/3\right)+\left((4\cdot 5)\cdot 6\right)-(7\cdot 8)+9.$$ (2.53)

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

$$\left(\left(((1\cdot 2)/3\right)+\left((4\cdot 5)\cdot 6)\right)-(7\cdot 8)\right)+9.$$

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

$$a\cdot b /c\cdot d/e\cdot f$$

meant

$$\left((a\cdot b)/(c\cdot d)\right)/(e\cdot f),$$

whereas today it means

$$\left(\left(\left((a\cdot b)/c\right)\cdot d\right)/e\right)\cdot f.$$

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

$$n\cdot n+1\cdot n+2\cdot n+3\cdot n+4$$

to mean

$$n\cdot (n+1)\cdot (n+2)\cdot (n+3)\cdot (n+4).$$

2.54   Examples. $\mbox{{\bf Q}}$ with the usual operations of addition and multiplication is a field.

$(\mbox{{\bf Z}}_5,\oplus_5,\odot_5)$ is a field. (See definition 2.42 for the definitions.) We showed in section 2.2 that $(\mbox{{\bf Z}}_5,\oplus_5,\odot_5)$ satisfies all the field axioms except possibly the distributive law. In appendix B, it is shown that the distributive property holds for $(\mbox{{\bf Z}}_n,\oplus_n,\odot_n)$ for all $n\in\mbox{{\bf N}}$, $n\geq 2$. (The proof assumes that the distributive law holds in $\mbox{{\bf Z}}$.)

For a general $n\in\mbox{{\bf N}}$, $n\geq 2$, the only field axiom that can possibly fail to hold in $(\mbox{{\bf Z}}_n,\oplus_n,\odot_n)$ is the existence of multiplicative inverses, so to determine whether $\mbox{{\bf Z}}_n$ is a field, it is just necessary to determine whether every non-zero element in $\mbox{{\bf Z}}_n$ is invertible for $\odot_n$.

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) $(\mbox{{\bf Z}},+,\cdot)$ where $+$ and $\cdot$ are usual addition and multiplication.
2. b) $(G,+,\cdot)$ where $G=\mbox{${\mbox{{\bf Q}}}^{+}$}\cup\{0\}$ is the set of non-negative rational numbers, and $+$ and $\cdot$ are the usual addition and multiplication.
3. c) $(H,+,\cdot)$ where $H=\{x\}$ is a set with just one element and both $+$ and $\cdot$ are the only binary operation on $H$; i.e.,
   
   $$x+x=x,\qquad x\cdot x=x.$$
   
   

4. d) $(\mbox{{\bf Q}},\oplus,\odot)$ where both $\oplus$ and $\odot$ are the usual operation of addition on $\mbox{{\bf Q}}$, e.g., $3\oplus 4 = 7$ and $3\odot 4=7$.

2.56   Exercise. Determine for which values of $n=2,3,4,5,6$, $(\mbox{{\bf Z}}_n,\oplus_n,\odot_n)$ is a field. (You already know that $n=5$ produces a field.)

2.57   Notation (The field $\mbox{{\bf Z}}_n$.) Let $n\in\mbox{{\bf N}}$, $n\geq 2$ be a number such that $(\mbox{{\bf Z}}_n,\oplus_n,\odot_n)$ is a field. Then `` the field $\mbox{{\bf Z}}_n$" means the field $(\mbox{{\bf Z}}_n,\oplus_n,\odot_n)$. I will often denote the operations in $\mbox{{\bf Z}}_n$ by $+$ and $\cdot$ instead of $\oplus_n$ and $\odot_n$.

2.58   Entertainment. Determine for which values of $n$ in $\{7,8,9,10,11\}$ the system $(\mbox{{\bf Z}}_n,\oplus_n,\odot_n)$ is a field. If you do this you will probably conjecture the exact (fairly simple) condition on $n$ that makes the system into a field.