06.5.ガロア群と対照群

ガロア群と対称群

Statement

$F:\text{Field}\quad(=\mathbb{Q})$
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
$\mathrm{Gal}(L_f/K) \simeq S_n$

Proof

$\sigma \in S_n$
$g(X_1,\cdots,X_n) \in K(X_1,\cdots,X_n)\quad$($\Leftarrow$ 有理関数体)
$(g(X_1,\cdots,X_n) = \frac{g'(X_1,\cdots,X_n)}{g''(X_1,\cdots,X_n)} \quad (g',g'' \in K[X_1,\cdots,X_n]))$
define $\sigma$ as
$\sigma: L_f \rightarrow L_f$
$\sigma: g(\alpha_1,\cdots,\alpha_n) \mapsto g(\alpha_{\sigma(1)},\cdots,\alpha_{\sigma(n)})$
then
$\sigma$:Automorphism over $K$
$\because k \in K, \sigma(k) = k\quad(\because K = F(\alpha_1,\cdots,\alpha_n))$
$\therefore \sigma \in \mathrm{Aut}(L_f/K) = \mathrm{Gal}(L_f/K)$
$\therefore S_n \subset \mathrm{Gal}(L_f/K)$

Let $M = L_f^{S_n}$
then $L_f/M$:Galois Extension
$\therefore [L_f:M] = |S_n| = n!\quad(\because\text{Prop.ガロアの基本定理(2)})$
$\therefore [L_f:K] = [L_f:M][M:K] \ge [L_f:M] = n!$---(1)

$F(\alpha_1, \cdots, \alpha_n) = K \subset K(\alpha_1) \subset \cdots K(\alpha_1, \cdots, \alpha_n) = L_f$
$[K(\alpha_1):K] \le n \quad(\because f(\alpha_1) = 0, \text{deg}f = n)$
$[K(\alpha_1,\alpha_2):K(\alpha_1)] \le n-1$
$(\because \text{ let } f(x) = (x - \alpha_1)f_2(x) \text{ then } f_2(x) \in K(\alpha_1)[x], f_2(\alpha_2) = 0, \text{deg}f_2 = n-1)$
$\quad\vdots$
$[K(\alpha_1,\cdots,\alpha_n):K(\alpha_1,\cdots,\alpha_{n-1})] \le 1$
$\therefore [L_f:K] = [K(\alpha_1,\cdots,\alpha_n):K] \le n!$---(2)

$(1),(2) \Rightarrow [L_f:K] = n!$
$n! = |S_n| \le |\mathrm{Gal}(L_f/K)| = |[L_f:K]| = n!$
$\therefore \mathrm{Gal}(L_f/K) = S_n$
$(\therefore M = K)$
QED.
$$ \begin{xy} \xymatrix { & L_f \ar@{-}[r] & \{id\} &\\ & M = L_f^{S_n} \ar[u]^{\text{(Galois)}} \ar@{-}[r] & S_n \ar[u] &\\ \ar[uu]^{\text{Galois}} & K=F(s_1,\cdots,s_n) \ar@{-}[u] \ar@{-}[r] & \text{Gal}(L_f/K) \ar@{-}[u] &\\ & F \ar@{-}[u] &\\ } \end{xy} $$