06.7.ガロア理論まとめ(代数方程式の可解条件まで)

ガロア理論まとめ(代数方程式の可解条件まで)

可解群

交換子:$[x,y] = xyx^{-1}y^{-1}$
交換子群:$D(G) = \langle \{[x,y]|x,y \in G\} \rangle$
可解群:$G:\text{ s.t. }\exists n \in \mathbb{N}, D^n (G) = \{e\}$

対称群の次数と可解性

$S_n$:n次対称群
$A_n$:n次交代群
$D(S_n) = A_n$
$5 \le n \Rightarrow D(A_n) = A_n$
$n \le 4 \Rightarrow S_n$:可解群
$5 \le n \Rightarrow S_n$:非可解群

有限可解群の同値条件

$G:\text{Finite Group}$
$\exists n \in \mathbb{N}, D^n(G) = \{e\}\quad(\overset{\text{def}}{\Leftrightarrow}\text{Solvable Group})$
$\Leftrightarrow$
$\exists (G_n) = (G = G_0, \supset \cdots \supset G_n = \{e\}) \\ \text{ s.t. } G_{i+1} \triangleleft G_i, G_i / G_{i+1}:\text{Cyclic Group}$

環準同型定理

$R,R'$:Ring
$f:R \mapsto R'$ is a homomorphism.
then
$R/\text{Ker }f \simeq \text{Img }f$
$$ \begin{xy} \xymatrix { & R / \ker f \ar@{-}[r]^{\simeq} & \text{Img} f \subset R'\\ & R \ar[u]_{\psi} \ar[ur]_{f} \\ } \end{xy} $$

極大イデアルと体

$R$:Ring
$I \varsubsetneqq R$:Ideal
Then
$I$:Maximal Ideal $\Leftrightarrow R/I$:Field

体の拡大

$F:\text{Field}$
$F(\alpha):\text{Field Extension}\quad \alpha \notin F$

代数拡大

$f(x) \in F[x], \text{ s.t. } f(\alpha) = 0, \text{deg}f = n, g(\alpha) = 0 \Rightarrow f(x)\mid g(x)$
$(f(x):\text{Minimal Polynomial of }\alpha)$ then
$F[x] / (f(x)) \simeq F[\alpha] = F(\alpha)$
$F(\alpha) = \{ c = b_{n-1} \alpha^{n-1} + \cdots + b_0\quad(b_{n-1},\cdots b_0 \in F)\}$
$[E:F] = n$
$$ \begin{xy} \xymatrix { & F[x] / (f(x)) \ar[r]^{\simeq} & F[\alpha] = F(\alpha)\\ F \ar[r] \ar[ur] & F[x] \ar[u]_{\psi} \ar[ur]_{\phi} \\ } \end{xy} $$

代数拡大の同型

$f(x) \in F[x]$
$f(x):\text{Minimal Polynomial}$
$f(x) = (x - \alpha_1)(x - \alpha_2)\cdots(x - \alpha_n)$
then $F(\alpha_i) \simeq F(\alpha_j) \quad(i \neq j)$
$\because F(\alpha_i) = F[x] / f(x) = F(\alpha_j)$

最小分解体

$f(x) \in F[x]$
$f(x) = (x - \alpha_1)\cdots(x - \alpha_n)$
$E = F(\alpha_1,\cdots,\alpha_n)$

体の同型の数

$F$:Field
$\alpha$:Algebraic element over $F$
$p(x)$:Minimal Polynomial of $\alpha$
$\mathbb{S} = \{\sigma:F(\alpha) \rightarrow \bar F|\sigma |_F = id_F,\sigma:\text{Homomorphism}\}$
then
$|\mathbb{S}| \le [F(\alpha):F]\quad(=\text{deg}p(x))$
and
$|\mathbb{S}| = [F(\alpha):F] \Leftrightarrow \alpha$:Separable over $F$

正規拡大

$E/F$:Finite Field Extension
then
$E/F$:Normal Extension $\Leftrightarrow \exists f(x) \in F[x]$ such that $E$:Minimal Splitting Field over $f(x)$

ガロア拡大

$E/F:\text{Galois} \overset{\text{def}}{\Leftrightarrow} E/F:\text{Separable and Normal}$

ガロアの基本定理

$E/F$:Finite Galois Extension
$G = \text{Gal}(E/F)$
(1)
$\text{Gal}(E/E^H) = H$
$E^{\text{Gal}(E/M)} = M$
(2)
(2.1)$M/F$:Galois Extension $\Leftrightarrow \text{Gal}(E/M) \triangleleft \text{Gal}(E/F)$
then
(2.2)$\text{Gal}(M/F) \simeq \text{Gal}(E/F)/\text{Gal}(E/M)$
$$ \begin{xy} \xymatrix { & E^{id} = E \ar@{-}[r] & \{id\} = \text{Gal}(E/E) &\\ & E^H = M \ar[u]^{\text{Galois}} \ar@{-}[r] & H = \text{Gal}(E/M) \ar@{-}[u] & \{id\} \ar@{-}[l]\\ \ar[uu]^{\text{Galois}} & E^G = F \ar@{-}[u]_{\text{(Galois)}} \ar@{-}[r] & G = \text{Gal}(E/F) \ar@{-}[u]_{(\triangleleft)} & \text{Gal}(E/F)/\text{Gal}(E/M) \simeq\text{Gal}(M/F) \ar@{-}[l] \ar@{-}[u] & \\ } \end{xy} $$

ガロアの推進定理の補題

$L,M$:Field
$E = L \cdot M \quad$(Composit Filed of $L$ and $M$)
$F = L \cap M$
$L/F$:Finite Galois Extension
$M/F$:Field Extension
then
$E/M$:Galois Extension
$\text{Gal}(E/M) \simeq \text{Gal}(L/F)$
$$ \begin{xy} \xymatrix { & E = L \cdot M & \\ L \ar[ur] & & M \ar[ul]_{\text{(Galois)}} & \\ & F = L \cap M \ar[ul]^{\text{Galois}} \ar[ur] & & \\ } \end{xy} $$

原始n乗根添加による拡大

$E = F(\sqrt[n]{1})$
$\overset{\text{(ii)}}{\Rightarrow}$
$E/F$:Cyclic Extension

巡回拡大と冪根拡大

$F$:Field
$p = \text{char} F$
if $p \gt 0$ then $p \nmid n$
$\sqrt[n]{1} \in F$
$\exists a \in F,E = F(\sqrt[n]{a})$
$\Leftrightarrow$
$E/F:\text{Cyclic Extension}\quad(\overset{\text{def}}{\Leftrightarrow} \text{Gal}(E/F):\text{Cyclic Group})$
$[E:F] \mid n$
$$ \begin{xy} \xymatrix { & E & \exists a \in F_1, E = F_1(\sqrt[n]{a}) \\ & F_1 = F_0(\sqrt[n]{1}) \ar[u] & \text{Gal}(E/F_1):\text{Cyclic} \ar@{-}[u]_{\Leftrightarrow} \\ & F_0 \ar[u]_{(\text{Cyclic})}\\ } \end{xy} $$

巡回拡大列と可解拡大

$E/F$:Galois Extension

$F = M_0 \subset M_1 \subset \cdots \subset M_n = E$
$M_{i+1}/M_i:\text{Cyclic Extension}\quad (\overset{\text{def}}\Leftrightarrow M_{i+1}/M_i:\text{Galois Extension},\text{Gal}(M_{i+1}/M_i):\text{Cyclic Group})$
$\Leftrightarrow$
$E/F$:Solvable Extension
$$ \begin{xy} \xymatrix { E & M_n \ar@{-}[r] & \{id\} = \text{Gal}(E/E) & & \\ & \vdots & \vdots & & \text{Gal}(E/M_{n}) / \text{Gal}(E/M_{n-1})\text{:Cyclic} \\ & M_{i+1} \ar@{-}[r] & \text{Gal}(E/M_{i+1}) & \{id\} & \vdots \\ & M_i \ar@{-}[u]^{\text{Cyclic}} \ar@{-}[r] & \text{Gal}(E/M_i) \ar@{-}[u]_{\triangleleft} & \text{Gal}(M_{i+1}/M_{i})\simeq \ar@{-}[u] & \text{Gal}(E/M_{i}) / \text{Gal}(E/M_{i+1})\text{:Cyclic} \\ & \vdots & \vdots & & \vdots\\ F \ar@{-}[uuuuu]^{\text{Galois}} & M_0 \ar@{-}[r] & \text{Gal}(E/F) & & \text{Gal}(E/M_{1}) / \text{Gal}(E/M_{0}) \text{:Cyclic} \\ } \end{xy} $$

冪根拡大と可解拡大

$F$: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)$E_f/F:\text{Radical Extension}(\overset{\text{def}}{\Leftrightarrow} \exists (F_i)_{i=0,\cdots m} \text{ s.t. } F_{i+1} = F_i(\sqrt[n_i]{a_i}), a_i \in F_i, F_0 = F, F_m \supset E_f)$
$\Leftrightarrow$
$E_f/F:\text{Solvable Extension}(\overset{\text{def}}{\Leftrightarrow} \text{Gal}(E_f/F):\text{Solvable Group}$) $(\Leftarrow)$
$$ \begin{xy} \xymatrix { & & E_f(\alpha_n) = M_m \ar@{-}[r] & H_m = \{id\} & & M_{m} = M_{m-1}(\sqrt[n_{m-1}]{b_{m-1}}) \\ \{id\} & E_f \ar@{-}[ur] & \vdots & \vdots \\ \vdots & \vdots & M_{i+1} \ar@{-}[r] & H_{i+1} & \{id\} & M_{i+1} = M_i(\sqrt[n_i]{b_i}) \\ & & M_i \ar@{-}[u]^{\text{(galois)}}\ar@{-}[r] & H_i \ar@{-}[u]_{\triangleleft} & H_i/H_{i+1}\text{:Cyclic} \ar@{-}[u] & \simeq \text{Gal}(M_{i+1}/M_i) \\ & & \vdots & \vdots \\ \vdots & \vdots & M_1 \ar@{-}[r] & H_1 & \{id\} & M_1 = M_0(\sqrt[n_0]{b_0})\\ & & F(\alpha_n) = M_0 \ar@{-}[u]^{\text{(galois)}}\ar@{-}[r] & H_0 = H \ar@{-}[u]_{\triangleleft} & H_0/H_1\text{:Cyclic} \ar@{-}[u] & \simeq \text{Gal}(M_1/M_0) \\ H \ar@{-}[u] & E_f \cap E_f(\alpha_n) \ar@{-}[u] \ar@{-}[ur] & & \\ G = \text{Gal}(E_f/F) \ar@{-}[u] & F \ar@{-}[u] \ar@{-}[uur]& } \end{xy} $$ $(\Rightarrow)$
$$ \begin{xy} \xymatrix { & & F_m' = F_{m-1}(\sqrt[n_{m-1}]{a_{m-1}}, \sqrt[n_{m-1}]{a_{m-1}^{(1)}}, \sqrt[n_{m-1}]{a_{m-1}^{(2)}},\cdots,, \sqrt[n_{m-1}]{a_{m-1}^{(s_{m-1})}}) & F_t''\\ & E_f \subset F_m = F_{m-1}(\sqrt[n_{m-1}]{a_{m-1}}) \ar@{-}[ur] & \vdots & \vdots \\ & \vdots & F_k'(\sqrt[n_k]{a_k}, \sqrt[n_k]{a_{k}^{(1)}}, \sqrt[n_k]{a_{k}^{(2)}},\cdots, \sqrt[n_k]{a_{k}^{(l)}}, \sqrt[n_k]{a_{k}^{(l+1)}}) & F_{i+1}'' \\ & \vdots & F_k'(\sqrt[n_k]{a_k}, \sqrt[n_k]{a_{k}^{(1)}}, \sqrt[n_k]{a_{k}^{(2)}},\cdots, \sqrt[n_k]{a_{k}^{(l)}}) \ar@{-}[u] & F_i'' \ar@{-}[u]_{\text{(Cyclic Extension)}} \\ & \vdots & \vdots \\ & \vdots & F_2' = F_1'(\sqrt[n_1]{a_1}, \sqrt[n_1]{a_{1}^{(1)}}, \sqrt[n_1]{a_{1}^{(2)}},\cdots,, \sqrt[n_1]{a_{1}^{(s_1)}}) & F''_3\\ & F_2 = F_1(\sqrt[n_1]{a_1}) & F_1' = F_0'(\sqrt[n_0]{a_0}) \ar@{-}[u] & F''_2 \ar@{-}[u]_{\text{(Cyclic Extension)}}\\ & F_1 = F_0(\sqrt[n_0]{a_0}) \ar@{-}[u] \ar@{-}[ur] & F_0' = F_0(\alpha_n) \ar@{-}[u]_{\text{(Cyclic Extension)}} & F''_1 \ar@{-}[u]_{\text{(Cyclic Extension)}}\\ & F_0 = F \ar@{-}[u] \ar@{-}[ur]_{\text{(Cyclic Ext.)}} \ar@{-}[uur]^{\text{(Galois Ext.)}} \ar@{-}[uuur]^{\text{(Galois Ext.)}} \ar@{-}[uuuuuuuur]^{\text{(Galois Ext.)}} & & F''_0 \ar@{-}[u]_{\text{(Cyclic Extension)}}\\ } \end{xy} $$

ガロア群と対照群

$F:\text{Field}\quad(=\mathbb{Q})$
char $(F) = 0$
$f(x) = (x - \alpha_1)\cdots(x - \alpha_n)$
$= x^n + s_1 x^{n-1} + \cdots + s_{n-1} x + s_n$
$(\alpha_1,\cdots,\alpha_n \notin F)$
$(s_1 = \alpha_1 + \cdots + \alpha_n, \cdots, s_n = \alpha_1\cdots\alpha_n)$
$K = F(s_1, \cdots, s_n)$
$f(x) \in K$:Minimal Polynomial over $K$
$L_f = K(\alpha_1, \cdots, \alpha_n) = F(\alpha_1, \cdots, \alpha_n)$
then
$\mathrm{Gal}(L_f/K) \simeq S_n$
$$ \begin{xy} \xymatrix { & L_f \ar@{-}[r] & \{id\} &\\ & M = L_f^{S_n} \ar[u]^{\text{(Galois)}} \ar@{-}[r] & S_n \ar[u] &\\ \ar[uu]^{\text{Galois}} & K=F(s_1,\cdots,s_n) \ar@{-}[u] \ar@{-}[r] & \text{Gal}(L_f/K) \ar@{-}[u] &\\ & F \ar@{-}[u] &\\ } \end{xy} $$

方程式の可解性

$\mathrm{Gal}(E_f/F) \simeq S_n$ (ガロア群と対照群)
$n \le 4 \Rightarrow S_n$:可解群 (対称群の次数と可解性)
$5 \le n \Rightarrow S_n$:非可解群 (対称群の次数と可解性)
$E_f/F:\text{Radical Extension}\Leftrightarrow :\text{Solvable Extension}$ (冪根拡大と可解拡大)