2. Number systems 3

2) For all x,y,t in F, we have (x+y)+t = x+(y +t) and (x*y)*t = x*(y *t)

(the associative laws for addition and multiplication).

3) There are distinct distinguished elements 0 and 1 in F such that, for all x

in F, we have 0 + x = x + 0 = x and 1 * x = x * 1 = x (the existence of additive

and multiplicative identities).

4) For each x in F and each y in F such that y 6 = 0, there are -x and

1

y

in F

such that x + (-x) = 0 and y *

1

y

= 1 (the existence of additive and multiplicative

inverses).

5) For all x,y,t in F we have t * (x + y) = (t * x) + (t * y) = t * x + t * y (the

distributive law).

For clarity and emphasis we repeat some of the main points. The rational

numbers provide a familiar example of a ﬁeld. In any ﬁeld we can add, subtract,

multiply, and divide as we expect, although we cannot divide by 0. The ability to

divide by a nonzero number distinguishes the rational numbers from the integers.

In more general settings the ability to divide by a nonzero number distinguishes

a ﬁeld from a commutative ring. Thus every ﬁeld is a commutative ring but a

commutative ring need not be a ﬁeld.

There are many elementary consequences of the ﬁeld axioms. It is easy to prove

that each element has a unique additive inverse and that each nonzero element has

a unique multiplicative inverse, or reciprocal. The proof, left to the reader, mimics

our early argument showing that subtraction is possible in Z.

Henceforth we will stop writing * for multiplication; the standard notation of

xy for x * y works adequately in most contexts. We also write x2 instead of xx as

usual. Let t be an element in a ﬁeld. We say that x is a square root of t if t = x2.

In a ﬁeld, taking square roots is not always possible. For example, we shall soon

prove that there is no rational square root of 2 and that there is no real square root

of -1.

At the risk of boring the reader we prove a few basic facts from the ﬁeld axioms;

the reader who wishes to get more quickly to geometric reasoning could omit the

proofs, although writing them out gives one some satisfaction.

Proposition 2.1. In a ﬁeld the following laws hold:

1) 0 + 0 = 0.

2) For all x, we have x0 = 0x = 0.

3) (-1)2 = (-1)(-1) = 1.

4) (-1)x = -x for all x.

5) If xy = 0 in F, then either x = 0 or y = 0.

Proof. Statement 1) follows from setting x = 0 in the axiom 0+x = x. Statement

2) uses statement 1) and the distributive law to write 0x = (0 + 0)x = 0x + 0x.

By property 4) of Deﬁnition 2.1, the object 0x has an additive inverse; we add this

inverse to both sides of the equation. Using the meaning of additive inverse and

then the associative law gives 0 = 0x. Hence x0 = 0x = 0 and 2) holds. Statement

3) is a bit more interesting. We have 0 = 1+(-1) by axiom 4) from Deﬁnition 2.1.