00.0.群環体の定義
群(Group)
$G$ is Set.Defined a mapping.
$$G \times G \longrightarrow G$$ $$a \in G, b \in G \quad (a,b) \mapsto a + b \in G $$ $G$ is Group. if $G$ fills following conditions.
1. $(a + b) + c = a + (b + c) \quad (a,b,c \in G)$
2. $\exists 0 \in G, \forall x \in G, 0 + x = x + 0 = x$
3. $\forall x \in G, \exists -x \in G, x + (-x) = -x + x = e$
可換群(Abelian Group)
$G$ is a Group.$\forall a,b \in G, a + b = b + a$
then $G$ is 可換群(アーベル群)
環(Ring)
$R$:可換群 $a,b,c \in R$* $(ab)c = a(bc)$
* $a(b + c) = ab + ac, (a + b)c = ac + bc$
* $\exists 1 \text{s.t.} \forall x \in R, x1 = 1x = x$
可換環(Ring)
$R$:環$a,b \in R$
* $ab = ba$
零因子
$R$:可換環 $a \in R$if
$\exists x \in R \text{ s.t. } x \neq 0, ax = 0$
then
$a$ is called 零因子
整域(Integral Domain)
$D$:零因子を持たない可換環体(Field)
$F$:可換環 $\forall x \in F - \{0\}, \exists x^{-1} \text{ s.t. } xx^{-1} = 1$部分群
$G$:群$H \subset G$ and $H$:群
部分環
$R$:環$S \subset R$ and $S$:環
部分体
$F$:体$E \subset F$ and $E$:体
($F$と同じ演算 ($+,\cdot$))