# Differences

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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> | ||

+ | )] |

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