# Fields

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### 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 : FG that is a homomorphism: h(x + y) = h(x) + h(y)  and  h(x·y) = h(xh(y)  and  h(1) = 1.

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

### Some results

0 is a zero for ·: x = x and x·0 = 0.

### Examples

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

### Properties

 Classtype first-order Equational theory Quasiequational theory First-order theory Locally finite no Residual size unbounded Congruence distributive yes Congruence modular yes Congruence n-permutable yes, n = 2 Congruence regular yes Congruence uniform yes Congruence extension property Definable principal congruences Equationally definable principal congruences Amalgamation property Strong amalgamation property Epimorphisms are surjective

### Finite members

[Size 1]?:  0
[Size 2]?:  1
[Size 3]?:  1
[Size 4]?:  1
[Size 5]?:  1
[Size 6]?:  0
There exists one field, called the Galois field GF(pm) of each prime-power order pm.

### Subclasses

[Fields of characteristic zero]?
[Algebraically closed fields]?

### Superclasses

Integral domains

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