06.4.冪根拡大と可解拡大

冪根拡大と可解拡大

Statement

$F$:Field
char $(F) = 0$
$f(x) \in F[x]$
$E_f$:Minimal Splitting Field of $f(x)$
then
(i)$E_f/F$:Galois Extension
and
(ii)$E_f/F$:Radical Extension $\Leftrightarrow E_f/F$:Solvable Extension($\Leftrightarrow \text{Gal}(E_f/F):\text{Solvable Group}$)

Proof

(i)
$\text{char}(F) = 0 \therefore E_f/F$:Separable Extension
$E_f$:Minimal Splitting Field of $f(x) \therefore E_f/F$:Normal Extension
$\therefore E_f/F:\text{Galois Extension} \quad(\because\text{Prop.ガロア拡大と最小分解体})$
(ii)
$\Leftarrow$
　 let
$G = \text{Gal}(E_F/F)$
$[E_f:F] = |G| = n$
$\alpha_n$:primitive n-th root(原始n乗根)
then
$E_f / E_f \cap F(\alpha_n)\text{:Galois Extension} \quad(\because E_f/F\text{:Galois}, E_F \cap F(\alpha_n) \supset F)$
$\therefore E_f(\alpha_n)/F(\alpha_n)\text{:Galois}\quad(\because\text{Prop.ガロアの推進定理})$
let
$H = \text{Gal}(E_f/E_f\cap F(\alpha_n))$
then
$H \subset G\quad(\because\text{Prop.ガロアの基本定理})$
$\therefore H:\text{Solvable Group}$
$H = \text{Gal}(E_f/E_f\cap F(\alpha_n)) \simeq \text{Gal}(E_f(\alpha_n)/F(\alpha_n))\quad(\because\text{Prop.ガロアの推進定理})$
$$ \begin{xy} \xymatrix { & & E_f \cdot F(\alpha_n) = E_f(\alpha_n) & & \{id\}\\ & E_f \ar[ur] & F(\alpha_n) \ar[u]_{\text{(Galois)}} & \{id\}\ar[ur] & H \ar[u] \\ & E_f \cap F(\alpha_n) \ar[u]^{\text{(Galois)}} \ar[ur] & & H \ar[u] \ar[ur] \\ \ar[uu]^{\text{Galois}} & F \ar[u] \ar[uur] & & G \ar[u] \\ } \end{xy} $$ $H\text{:Finite Solvable Group}\quad(\because\text{Prop.可解群の基本性質(i)})$
$\exists (H_m) = (H = H_0, \supset \cdots \supset H_m = \{id\}) \\ \text{ s.t. } H_{i+1} \triangleleft H_i, H_i / H_{i+1}:\text{Cyclic Group}\quad(\because\text{Prop.有限可解群の同値条件})$
let
$M_i = E_f(\alpha_n)^{H_i}\quad(\because\text{Prop.ガロアの基本定理}(2))$
then
$M_m/M_i:\text{Galois Extension}\quad(\because M_m/M_0:\text{Galois Extension})$
$H_{i+1} \triangleleft H_i$
$M_{i+1}/M_i:\text{Galois Extension}\quad(\because\text{Prop.ガロアの基本定理(6-1)})$ $\text{Gal}(M_{i+1}/M_i) \simeq H_i/H_{i+1}\quad(\because\text{Prop.ガロアの基本定理(6-2)})$
$\therefore \text{Gal}(M_{i+1}/M_i) \text{:Cyclic Group}$
$\therefore M_{i+1}/M_i:\text{Cyclic Extension}$
$$ \begin{xy} \xymatrix { & & E_f(\alpha_n) = M_m \ar@{-}[r] & H_m = \{id\} & & M_{m} = M_{m-1}(\sqrt[n_{m-1}]{b_{m-1}}) \\ \{id\} & E_f \ar@{-}[ur] & \vdots & \vdots \\ \vdots & \vdots & M_{i+1} \ar@{-}[r] & H_{i+1} & \{id\} & M_{i+1} = M_i(\sqrt[n_i]{b_i}) \\ & & M_i \ar@{-}[u]^{\text{(galois)}}\ar@{-}[r] & H_i \ar@{-}[u]_{\triangleleft} & H_i/H_{i+1}\text{:Cyclic} \ar@{-}[u] & \simeq \text{Gal}(M_{i+1}/M_i) \\ & & \vdots & \vdots \\ \vdots & \vdots & M_1 \ar@{-}[r] & H_1 & \{id\} & M_1 = M_0(\sqrt[n_0]{b_0})\\ & & F(\alpha_n) = M_0 \ar@{-}[u]^{\text{(galois)}}\ar@{-}[r] & H_0 = H \ar@{-}[u]_{\triangleleft} & H_0/H_1\text{:Cyclic} \ar@{-}[u] & \simeq \text{Gal}(M_1/M_0) \\ H = \text{Gal}(E_f/E_f \cap F(\alpha_n)) \ar@{-}[u] & E_f \cap F_f(\alpha_n) \ar@{-}[u] \ar@{-}[ur] & & \\ G = \text{Gal}(E_f/F) \ar@{-}[u] & F \ar@{-}[u] \ar@{-}[uur]& } \end{xy} $$ let
$n_i = [M_{i+1}:M_i]$
$h_i = |H_i|$
then
$n_i | n \quad(\because n_i = [M_{i+1}:M_i] = |\text{Gal}(M_{i+1}/M_i)| = |H_i/H_{i+1}|, \therefore n_i h_{i+1}=h_i , h_i|h_0, h_0|n)$
$\therefore \exists d_i \in \mathbb{N} \text{ s.t. } n_i d_i = n$
$\therefore 1 = \alpha_n^n = \alpha_n^{n_i d_i} = (\alpha_n^{d_i})^{n_i}$ $\therefore \alpha_n^{d_i}$:primitive $n_i$-th root
$\alpha_n^{d_i} \in M_i \quad(\because \alpha_n \in M_0 \subset M_i)$
$M_{i+1}/M_i:\text{Cyclic Extension}$
$\therefore\exists b_i \in M_i \text{ s.t. } M_{i+1} = M_i(\sqrt[n_i]{b_i}) \quad(\because\text{Prop.巡回拡大(iii)})$
$E_f \subset E_f(\alpha_n) = M_m = M_{m-1}(\sqrt[n_{m-1}]{b_{m-1}}) = \cdots = M_0(\sqrt[n_0]{b_0},\cdots,\sqrt[n_{m-1}]{b_{m-1}}) = F(\alpha_n = \sqrt[n]{1},\sqrt[n_0]{b_0},\cdots,\sqrt[n_{m-1}]{b_{m-1}})$ $\therefore E_f/F$:Radical Extension

$\Rightarrow$
$E_f/F$:Radical Extension
$\therefore \exists (F_i)_{i=0,\cdots m} \text{ s.t. } F_{i+1} = F_i(\sqrt[n_i]{a_i}), a_i \in F_i, F_0 = F, F_m \supset E_f$
let
$n = n_0 n_1 \cdots n_{m-1}$
$\alpha_n = \sqrt[n]{1}$(primitive n-th root)
$$ \begin{xy} \xymatrix { & & F_m' = F_{m-1}(\sqrt[n_{m-1}]{a_{m-1}}, \sqrt[n_{m-1}]{a_{m-1}^{(1)}}, \sqrt[n_{m-1}]{a_{m-1}^{(2)}},\cdots,, \sqrt[n_{m-1}]{a_{m-1}^{(s_{m-1})}}) & F_t''\\ & E_f \subset F_m = F_{m-1}(\sqrt[n_{m-1}]{a_{m-1}}) \ar@{-}[ur] & \vdots & \vdots \\ & \vdots & F_k'(\sqrt[n_k]{a_k}, \sqrt[n_k]{a_{k}^{(1)}}, \sqrt[n_k]{a_{k}^{(2)}},\cdots, \sqrt[n_k]{a_{k}^{(l)}}, \sqrt[n_k]{a_{k}^{(l+1)}}) & F_{i+1}'' \\ & \vdots & F_k'(\sqrt[n_k]{a_k}, \sqrt[n_k]{a_{k}^{(1)}}, \sqrt[n_k]{a_{k}^{(2)}},\cdots, \sqrt[n_k]{a_{k}^{(l)}}) \ar@{-}[u] & F_i'' \ar@{-}[u]_{\text{(Cyclic Extension)}} \\ & \vdots & \vdots \\ & \vdots & F_2' = F_1'(\sqrt[n_1]{a_1}, \sqrt[n_1]{a_{1}^{(1)}}, \sqrt[n_1]{a_{1}^{(2)}},\cdots,, \sqrt[n_1]{a_{1}^{(s_1)}}) & F''_3\\ & F_2 = F_1(\sqrt[n_1]{a_1}) & F_1' = F_0'(\sqrt[n_0]{a_0}) \ar@{-}[u] & F''_2 \ar@{-}[u]_{\text{(Cyclic Extension)}}\\ & F_1 = F_0(\sqrt[n_0]{a_0}) \ar@{-}[u] \ar@{-}[ur] & F_0' = F_0(\alpha_n) \ar@{-}[u]_{\text{(Cyclic Extension)}} & F''_1 \ar@{-}[u]_{\text{(Cyclic Extension)}}\\ & F_0 = F \ar@{-}[u] \ar@{-}[ur]_{\text{(Cyclic Ext.)}} \ar@{-}[uur]^{\text{(Galois Ext.)}} \ar@{-}[uuur]^{\text{(Galois Ext.)}} \ar@{-}[uuuuuuuur]^{\text{(Galois Ext.)}} & & F''_0 \ar@{-}[u]_{\text{(Cyclic Extension)}}\\ } \end{xy} $$ then
let $F'_0 = F_0(\alpha_n)$ then $F_0'/F_0:\text{Cyclic Extension}\quad(\because\text{Prop.原始n乗根添加による拡大})$
let $F'_1 = F'_0(\sqrt[n_0]{a_0})$ then $F'_1/F'_0:\text{Cyclic Extension}\quad(\because\text{Prop.巡回拡大})$
then
$F'_1:\text{Minimal Splitting Field of }(x^{n_0} - a_0) \in F[x]$
$F'_1/F_0$:Galois Extension
$i \ge 1$
$\exists f_i(x):\text{Minimal Polynomial of } a_i \in F[x] \quad(f_i(a_i) = 0) \quad(\because F_i/F:\text{Algebraic Extension})$
let
$f_1(x) = (x - a_1)(x - a_1^{(2)})\cdots(x - a_1^{(s_1)})\quad(a_1=a_1^{(1)}, a_1^{(2)}, \cdots, a_1^{(s_1)} \in \bar F)$
$f_2(x) = (x - a_2)(x - a_2^{(2)})\cdots(x - a_2^{(s_1)})\quad(a_2=a_2^{(1)}, a_2^{(2)}, \cdots, a_2^{(s_2)} \in \bar F)$
$\vdots$
$f_m(x) = (x - a_m)(x - a_m^{(2)})\cdots(x - a_m^{(s_m)})\quad(a_m=a_m^{(1)}, a_m^{(2)}, \cdots, a_m^{(s_m)} \in \bar F)$

$f'_0(x) = x^{n_0} - a_0 \in F[x]$
$f'_1(x) = f_1(x^{n_1}) = (x^{n_1} - a_1^{(1)})\cdots(x^{n_1} - a_1^{(s_1)}) \in F[x]$
$f'_2(x) = f_2(x^{n_2}) = (x^{n_2} - a_2^{(1)})\cdots(x^{n_2} - a_2^{(s_2)}) \in F[x]$
$\vdots$
$f'_m(x) = f_m(x^{n_m}) = (x^{n_m} - a_m^{(1)})\cdots(x^{n_m} - a_m^{(s_m)})$

then
$a_1=a_1^{(1)} \in F_1, a_1^{(2)}, \cdots, a_1^{(s_1)} \in F'_1\quad(\because F'_1/F_0:\text{Normal Extension})$
let $F'_2 = F'_1(\sqrt[n_1]{a_1^{(1)}}, \cdots, \sqrt[n_1]{a_1^{(s_1)}})$
then $F'_2/F_0:\text{Minimal Splitting Field of }f'_0(x)f'_1(x) \in F[x]$
$\therefore\text{Normal Extension}\therefore\text{Galois Extension}$

$a_2=a_2^{(1)} \in F_2, a_2^{(2)}, \cdots, a_2^{(s_2)} \in F'_2\quad(\because F'_2/F_0:\text{Normal Extension})$
let $F'_3 = F'_2(\sqrt[n_2]{a_2^{(1)}}, \cdots, \sqrt[n_2]{a_2^{(s_2)}})$
then $F'_3/F_0:\text{Minimal Splitting Field of }f'_0(x)f'_1(x)f'_2(x) \in F[x]$
$\therefore\text{Normal Extension}\therefore\text{Galois Extension}$
$\vdots$

$a_{m-1}=a_{m-1}^{(1)} \in F_{m-1}, a_{m-1}^{(2)}, \cdots, a_{m-1}^{(s_{m-1})} \in F'_{m-1}\quad(\because F'_{m-1}/F_0:\text{Normal Extension})$
let $F'_m = F'_{m-1}(\sqrt[n_{m-1}]{a_{m-1}^{(1)}}, \cdots, \sqrt[n_{m-1}]{a_{m-1}^{(s_{m-1})}})$
then $F'_m/F_0:\text{Minimal Splitting Field of }\prod_{i=0}^{m-1} f'_i(x) \in F[x]$
$\therefore\text{Normal Extension}\therefore\text{Galois Extension}$
$F''_0 = F_0 = F$
$F''_1 = F'_0 = F_0(\alpha_n)$
$F''_2 = F'_1 = F'_0(\sqrt[n_0]{a_0}) \quad(a_0 \in F_0 \subset F'_0 = F''_1)$
$F''_3 = F_1^{(1)} = F'_1(\sqrt[n_1]{a_1^{(1)}}) \quad(a_1^{(1)} = a_1 \in F_1 \subset F'_1 = F''_2))$
$F''_4 = F_1^{(2)} = F'_1(\sqrt[n_1]{a_1^{(1)}}, \sqrt[n_1]{a_1^{(2)}}) \quad(a_1^{(2)} \in F'_1 \subset F''_3))$
$\vdots$
$F''_{2+s_1} = F'_2 = F_1^{(s_1)} = F'_1(\sqrt[n_1]{a_1^{(1)}}, \cdots, \sqrt[n_1]{a_1^{(s_1)}}) \quad(a_1^{(s_1)} \in F'_1 \subset F''_{2 + s_1 - 1})$
$F''_{2+s_1+1} = F_2^{(1)} = F'_2(\sqrt[n_2]{a_2^{(1)}}) \quad(a_2^{(1)} = a_2 \in F_2 \subset F'_2 = F''_{2+s_1})$
$\vdots$
$F''_{2+s_1+s_2} = F'_3 = F_2^{(s_2)} = F'_2(\sqrt[n_2]{a_2^{(1)}}, \cdots, \sqrt[n_2]{a_2^{(s_2)}}) \quad(a_2^{(s_2)} \in F'_2 \subset F''_{2 + s_1 + s_2 - 1})$
$\vdots$
$F''_t = F'_m = F_{m-1}^{(s_{m-1})} = F'_{m-1}( \sqrt[n_{m-1}]{a_{m-1}^{(1)}}, \cdots, \sqrt[n_{m-1}]{a_{m-1}^{(s_{m-1})}}) \quad(a_{m-1}^{(s_{m-1})} \in F'_{m-1} \subset F''_{2 + s_1 + \cdots + s_{m-1} - 1})$
$(t=2 + s_1 + \cdots + s_{m-1})$ $F''_{i+1}/F''_i:\text{Cyclic Extension}\quad (0 \le i \lt t)\quad(\because\text{Prop.巡回拡大と冪根拡大})$
$\therefore F''_t/F''_0:\text{Solvable Extension}\quad(\because\text{Prop.巡回拡大列と可解拡大})$
$\therefore F'_m/F:\text{Solvable Extension}$
$\therefore \text{Gal}(F'_m/F):\text{Solvable Group}$
$F \subset E_f \subset F'_m$
$E_f/F:\text{Galois Extension}\quad(\because\text{Prop.代数方程式の可解条件(1)})$
$\therefore \text{Gal}(F'_m/E_f) \triangleleft \text{Gal}(F'_m/F)\quad(\because\text{Prop.ガロアの基本定理(6-1)})$
$\text{Gal}(E_f/F) \simeq \text{Gal}(F'_m/F)/\text{Gal}(F'_m/E_f)) \quad(\because\text{Prop.ガロアの基本定理(6-2)})$
$\text{Gal}(F'_m/F)/\text{Gal}(F'_m/E_f)):\text{Solvable Group}\quad(\because\text{Prop.交換子群の基本性質(ii)})$
$\therefore\text{Gal}(E_f/F):\text{Solvable Group}$
$E_f/F:\text{Solvable Extension}$
QED.