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
such that x + (-x) = 0 and y *
= 1 (the existence of additive and multiplicative
5) For all x,y,t in F we have t * (x + y) = (t * x) + (t * y) = t * x + t * y (the
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
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.