Differences

—		fields [2010/07/29 15:46] (current)
Line 1:		Line 1:
		+	=====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>
		+	)]