06.6.代数方程式の可解条件

代数方程式の可解条件

Statement

$F$:Field
char $(F) = 0$
$f(x) \in F[x]$:Minimal Polynomial over $F$
$n =$ degree of $f(x)$
then
$f(x)$ is solvable if and only if $n \le 4$

Proof

let
$f(x) = (x - \alpha_1)\cdots(x - \alpha_n)$
$L_f = F(\alpha_1, \cdots, \alpha_n)$
then
$L_f$:Minimal Splitting Field of $f(x)$
$L_f/F:\text{Separable Extension}\quad(\because\text{Prop.標数0の体の代数拡大は分離拡大})$
$\therefore L_f/F$:Galois Extension
$\text{Gal}(L_f/F) = S_n \quad(\because\text{Prop.ガロア群と対称群})$
when $n \le 4$ then
$S_n:\text{Solvable Group}\quad(\because\text{Prop.対称群の次数と可解群})$
$\therefore \text{Gal}(L_f/F):\text{Solvable}$
$\therefore L_f/F:\text{Solvable Extension}$
$\therefore\text{Radical Extension}\quad(\because\text{Prop.冪根拡大と可解拡大})$
when $5 \ge n$ then
$S_n:\text{not-Solvable Group}\quad(\because\text{Prop.対称群の次数と可解群})$
$\therefore \text{Gal}(L_f/F):\text{not-Solvable}$
$\therefore L_f/F:\text{not-Solvable Extension}$
$\therefore\text{not-Radical Extension}\quad(\because\text{Prop.冪根拡大と可解拡大})$