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fields [2010/07/29 15:46] (current)
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 +=====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>
 +)]