05.3.分離拡大

分離拡大の同値条件

Statement

$E/F$:Finite Extension
$\mathbb{S} = \{\sigma:E \rightarrow \bar F, \sigma|_F = id\}$
then
$E/F$:Separable Extension $\Leftrightarrow [E:F] = |\mathbb{S}|$

Proof

($\Rightarrow$)
$E/F$:Finite Separable Extension $\Rightarrow E/F$:Simple Extension $\quad(\because\text{Prop.有限次分離拡大は単拡大-2})$
$E/F$:Simple Separable Extension $\Rightarrow [E:F] = |\mathbb{S}|\quad(\because\text{Prop.単拡大と同型写像})$
($\Leftarrow$)
$\forall \alpha \in E$
let
$f(x) \in F[x]$:Minimal Polynomial of $\alpha$ over $F$
$\mathbb{M} = \{\mu:F(\alpha) \rightarrow \bar F, \mu|_F = id\}$
$\mathbb{T} = \{\tau:E \rightarrow \bar F, \tau|_{F(\alpha)} = id\}$
then
$|\mathbb{M}| \le [F(\alpha):F]\quad(\because\text{Prop.有限次拡大と同型写像})$
$|\mathbb{T}| \le [E:F(\alpha)]\quad(\because\text{Prop.有限次拡大と同型写像})$
Suppose $|\mathbb{M}| \lt [F(\alpha):F]$
then $|\mathbb{S}| = |\mathbb{T}||\mathbb{M}| \lt [E:F(\alpha)][F(\alpha):F] = [E:F]$
$\therefore |\mathbb{M}| = [F(\alpha):F]$
$\therefore \alpha$:Separable over $F \quad(\because\text{Prop.単拡大と同型写像})$
$\therefore E/F$:Separable Extension
QED

中間体と分離拡大

Statement

$E/M/F$:Algebraic Extension
then
$E/F$:Separable Extension $\Leftrightarrow E/M,M/F$:Separable Extension

Proof

($\Rightarrow$)
$\forall \alpha \in M \subset E$
$\therefore \alpha$:Separable
$\therefore M/F$:Separable

$\forall \alpha \in E$
$p(x) \in M[x]$:Minimal Polynomial
$q(x) \in F[x]$:Minimal Polynomial
$p(x) | q(x)$
$q(x)$:Separable Polynomial
$\therefore p(x)$:Separable
$\therefore \alpha$:Separable
$\therefore E/M$:Separable
($\Leftarrow$)
$\forall \alpha \in E$
let
$f(x) \in M[x]$:Minimal Polynomial
$f(x) = x^n + m_{n-1}x^{n-1} + \cdots + m_0 \quad(m_{n-1},\cdots,m_0 \in M)$
$L = F(m_0,\cdots,m_{n-1})$
then
$f(x) \in L[x]$
$f(\alpha) = 0$
$f(x)$:Separable
$F \subset L \subset M \subset E$
$L/F$:Separable Extension $\because M/F$:Separable Extension
$L(\alpha)/L$:Separable　$\therefore |\text{Hom}_L L(\alpha)| = [L(\alpha):L]$
$L/F$:Separable　$\therefore |\text{Hom}_F L| = [L:F]$
$|\text{Hom}_F L(\alpha)| = |\text{Hom}_L L(\alpha)||\text{Hom}_F L|\quad(\because\text{Prop.中間体の同型写像の数 })$ $[L(\alpha):F] = [L(\alpha):L][L:F] \quad(\because\text{Prop.ラグランジェの定理})$
$\therefore|\text{Hom}_F L(\alpha)| = [L(\alpha):F]$
$\therefore L(\alpha)/F$:Separable $\quad(\because\text{Prop.分離拡大の同値条件})$
$\therefore \alpha$:Separable over $F$
QED
その２
$E/M$:Separable　$\therefore |\text{Hom}_M E| = [E:M]$
$M/F$:Separable　$\therefore |\text{Hom}_F M| = [M:F]$
$\therefore$
$|\text{Hom}_F E| = |\text{Hom}_F M||\text{Hom}_M E| \quad(\because\text{Prop.中間体の同型写像の数})$ $[E:F] = [E:M][M:F] \quad(\because\text{Prop.ラグランジェの定理})$
$\therefore|\text{Hom}_F E| = [E:F]$
$\therefore E/F$:Separable $\quad(\because\text{Prop.分離拡大の同値条件})$
QED

分離的元の添加

Statement

$E/F$:Algebraic Field Extension
$S \subset E$
$\forall s \in S, s$:Separable
then
$F(S)/F$:Separable Extension

Proof

$[F(S):F] \le [E:F(S)][F(S):F] = [E:F] \lt \infty$
$\therefore \exists \{s_1,\cdots,s_n\}, F(S) = F(\{s_1,\cdots,s_n\})$
$[F(S):F] = [F(S):F(\{s_1,\cdots,s_{n-1})]\cdots[F(s_1):F]$
let
$F(s_i,\cdots,s_1) = F_i, F(s_0) = F_0$ and
$\phi_i:F_i \rightarrow \bar F, x \in F_{i-1}, \phi(x) = x$
$\forall i, [F_i:F_{i-1}] = |\phi_i|$
$\therefore F_i/F_{i-1}:\text{Separable}\quad(\because\text{Prop.分離拡大の同値条件})$
$\therefore F(S)/F:\text{Separable}\quad(\because\text{Prop.中間体と分離拡大})$