=====Fields=====

Abbreviation: **Fld**
====Definition====
A \emph{field} is a [[commutative rings with identity]] $\mathbf{F}=\langle F,+,-,0,\cdot,1
\rangle$ such that


$\mathbf{F}$ is non-trivial:  $0\ne 1$


every non-zero element has a multiplicative inverse:  $x\ne 0\Longrightarrow \exists y 
(x\cdot y=1)$

Remark: 
The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.


==Morphisms==
Let $\mathbf{F}$ and $\mathbf{G}$ be fields. A morphism from $\mathbf{F}$
to $\mathbf{G}$ is a function $h:F\rightarrow G$ that is a homomorphism: 

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

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

====Examples====
Example 1: $\langle\mathbb{Q},+,-,0,\cdot,1\rangle$, the field of rational numbers with addition, subtraction, zero, multiplication, and one.


====Basic results====
$0$ is a zero for $\cdot$: $0\cdot x=x$ and $x\cdot 0=0$.

====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 def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &0\\
f(2)= &1\\
f(3)= &1\\
f(4)= &1\\
f(5)= &1\\
f(6)= &0\\
\end{array}$

There exists one field, called the Galois field $GF(p^m)$ of each prime-power order $p^m$.

====Subclasses====
[[Fields of characteristic zero]] 

[[Algebraically closed fields]] 

====Superclasses====
[[Integral domains]] 


====References====

[(Ln19xx>
)]