Fields

Abbreviation: Fld

Definition

A field is a commutative ring with identity F = (F, + ,−,0,·,1 ) such that
F is non-trivial:   0 ≠ 1 and
every non-zero element has a multiplicative inverse:   x ≠ 0  ⇒  ∃y (x·y = 1).

Remark: The inverse of x is unique, and is usually denoted by x⁻¹.

Morphisms

Let F and G be fields. A morphism from F to G is a function h : F→G that is a homomorphism: h(x + y) = h(x) + h(y)  and  h(x·y) = h(x)·h(y)  and  h(1) = 1.

Remark: It follows that h(0) = 0 and h(−x) = −h(x).

Some results

0 is a zero for ·: 0·x = x and x·0 = 0.

Examples

(Q, + ,−,0,·,1), the field of rational numbers with addition, subtraction, zero, multiplication, and one.

Properties

Finite members

Subclasses

Superclasses