1.1. Definitions. A field is a set F with two composition laws + and ・ such that

Back

(a) (F,+) is an abelian group;

(b) let F× = F − {0};the n (F×, ・) is an abelian group;

(c) (distributive law) for all a, b, c ∈ F, (a + b)c = ac + bc (hence also a(b + c) = ab + ac).

Equivalently, a field is a nonzero commutative ring (meaning with 1) such that every nonzero

element has an inverse. A field contains at least two distinct elements, 0 and 1. The smallest,

and one of the most important, fields is F2 = Z/2Z = {0, 1}.

Lemma 1.1. A commutative ring R is a field if and only if it has no ideals other than (0)

and R.

Proof. Suppose R is a field, and let I be a nonzero ideal in R. If a is a nonzero element

of I, then 1 = a−1a ∈ I, and so I = R. Conversely, suppose R is a commutative ring with

no nontrivial ideals;if a _= 0, then (a) = R, which means that there is a b in F such that

ab = 1.

Example 1.2. The following are fields: Q, R, C, Fp = Z/pZ.

A homomorphism of fields α : F → F_ is simply a homomorphism of rings, i.e., it is a map

with the properties

α(a + b) = α(a) + α(b), α(ab) = α(a)α(b), α(1) = 1, all a, b ∈ F.

Such a homomorphism is always injective, because the kernel is a proper ideal (it doesn’t

contain 1), which must therefore be zero.

Навигация