06.2.ガロアの推進定理

ガロア群の中間体への制限

Statement

$E/M$:Galois Extension
$M/F$:Galois Extension
$\sigma \in \text{Gal}(E/F)$
$H = \{\sigma|_M\}$
$G = \{$
$\psi(\rho,\mu):E \rightarrow E$ such that
$x \in E-M, \psi(x) = \rho(x) \quad(\rho \in \text{Gal}(E/M))$
$x \in M, \psi(x) = \mu(x) \quad(\mu \in \text{Gal}(M/F))$
$\}$
($G \simeq \text{Gal}(E/M) \times \text{Gal}(M/F)$)
then
$G = \text{Gal}(E/F)$
$H = \text{Gal}(M/F)$
$\sigma(M) = M$

Proof

$\forall \psi \in G$
$\psi$:Homomorphism
$\psi(x) = x \quad (x \in F)$
$\therefore \psi \in \text{Gal}(E/F)$
$\therefore G \subset \text{Gal}(E/F)$
$|G| = |\text{Gal}(E/M) \times \text{Gal}(M/F)| = |\text{Gal}(E/M)||\text{Gal}(M/F)| = [E:M][M:F] = [E:F] = |\text{Gal}(E/F)|$
$\therefore G = \text{Gal}(E/F)$
$\therefore H = \text{Gal}(M/F)$
$\therefore \sigma(M) = M$
$$ \begin{xy} \xymatrix { E \ar[r]^{\sigma} & E \ar[r]^{\rho} & E & E \\ M \ar@{-}[u] & M \ar[r]^{id} \ar@{-}[u] & M \ar[r]^{\mu} \ar@{-}[u] & M \\ F \ar@{-}[u] \ar[r]^{id} & F \ar@{-}[u] & F \ar[r]^{id} \ar@{-}[u] & F \ar@{-}[u] \\ } \end{xy} $$ QED.

ガロアの推進定理の補題

Statement

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

Proof

(i)
$\exists f(x) \in F[x] \text{ s.t. } f(x) = \prod_{i=1}^{n}(x - \alpha_i), L = F(\alpha_1, \cdots, \alpha_n) \quad(\alpha_i \neq \alpha_j) \quad(\because\text{Prop.ガロア拡大と最小分解体})$
$\therefore E = L \cdot M = M(L) = M(F(\alpha_1,\cdots,\alpha_n)) = M(\alpha_1,\cdots,\alpha_n) \quad(\because F \subset M)$
$f(x) \in M[x] \quad(\because F[x] \subset M[x])$
$f(x)$:Separable
$\therefore E/M:\text{Galois Extension} \quad(\because\text{Prop.ガロア拡大と最小分解体})$
Define $\phi$ as
$\phi:\text{Gal}(E/M) \rightarrow \text{Gal}(L/F)$
$\phi:\sigma \in \text{Gal}(E/M) \mapsto \sigma|_L \in \text{Gal}(L/F)$
then
for $\forall \sigma \in \text{Gal}(E/M)$
$\sigma(L) \subset L$
($\because$
$L = F(\alpha_i, \cdots, \alpha_n)$
$\sigma(F) = id \because F \subset M, \sigma(M) = id$
$\sigma(\alpha_i) = \alpha_j \quad(\because \sigma \in \text{Gal}(E/M), E = M(\alpha_1, \cdots, \alpha_n))$
)
$\therefore \phi$:well-defined
$\phi:\text{Homomorphism} \quad(\because \sigma\tau |_L = \sigma|_L \tau|_L)$
$\phi:\text{Injection} \quad(\because \phi(\sigma) = id \Rightarrow \sigma|_L = id , \sigma|_M = id \therefore \sigma = id)$
$\phi$:Surjection
($\because$
$L^{\phi(\text{Gal}(E/M))} = \{x \in L|\forall \sigma \in \text{Gal}(E/M), \phi(\sigma)(x) = \sigma|_L(x) = x\}$
$\subset\{x \in E|\forall \sigma \in \text{Gal}(E/M), \sigma(x) = x\}$
$= E^{\text{Gal}(E/M)} = M$
$\therefore$ $L^{\phi(\text{Gal}(E/M))} \subset L \cap M = F$

$x \in L \cap M, \forall \sigma \in \text{Gal}(E/M)$
$\phi(\sigma)(x) = \sigma|_L(x) = \sigma(x) = x$
$\therefore x \in L^{\phi(\text{Gal}(E/M))}$
$L^{\phi(\text{Gal}(E/M))} \supset L \cap M = F$

$\therefore L^{\phi(\text{Gal}(E/M))}=L \cap M = F$
$\therefore \text{Gal}(L/F) = \phi(\text{Gal}(E/M)) \quad(\because\text{Prop.ガロアの基本定理(2)(3)})$
)

$\therefore \text{Gal}(L/F) \simeq \text{Gal}(E/M)$
(ii)
$E/M$:Galois Extension $\quad(\because\text{(i)})$
$E/L$:Galois Extension $\quad(\because\text{(i)})$
$\text{Gal}(L/F) \simeq \text{Gal}(E/M)\quad(\because\text{(i)})$
$\text{Gal}(M/F) \simeq \text{Gal}(E/L)\quad(\because\text{(i)})$
Define $\phi$ as
$\phi:\text{Gal}(E/F) \rightarrow \text{Gal}(L/F) \times \text{Gal}(M/F)$
$\phi:\sigma \mapsto (\sigma|_L, \sigma|_M)$
then
$\phi:\text{well-defined}\quad(\because\text{Prop.ガロア群の中間体への制限})$
$\phi$:Homomorphism
$\because \phi(\sigma\tau) = \phi(\sigma)\phi(\tau)$
$\phi$:Injective
$\because \phi(\sigma) = id \rightarrow (\sigma|_L, \sigma|_M) = id \Rightarrow \sigma|_L = id_L, \sigma|_M = id_m \therefore \sigma = id$
$\phi$:Surjective
$\because |\text{Gal}(E/F)| = [E:F]$
$= [E:L][L:F]$
$= |\text{Gal}(E/L)||\text{Gal}(L/F)|$
$= |\text{Gal}(M/F)||\text{Gal}(L/F)|\quad(\because\text{(i)})$
$= |\text{Gal}(M/F) \times \text{Gal}(L/F)|$

$\therefore\text{Gal}(E/F) \simeq \text{Gal}(L/F) \times \text{Gal}(M/F)$

ガロアの推進定理

Statement

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

Proof

(i)
$L/F':\text{Galois Extension} \quad(\because L/F\text{:Galois Extension and } F' \subset L)$
then
$E/M:\text{Galois Extension}\quad(\because\text{Prop.ガロアの推進定理の補題(i)})$
$\text{Gal}(E/M) \simeq \text{Gal}(L/F')\quad(\because\text{Prop.ガロアの推進定理の補題(i)})$
(ii)
$L/F':\text{Galois Extension} \quad(\because L/F\text{:Galois Extension and } F' \subset L)$
$M/F':\text{Galois Extension} \quad(\because M/F\text{:Galois Extension and } F' \subset L)$
then
$E/M:\text{Galois Extension}\quad(\because\text{Prop.ガロアの推進定理の補題(i)})$
$\text{Gal}(E/M) \simeq \text{Gal}(L/F')\quad(\because\text{Prop.ガロアの推進定理の補題(i)})$
$E/L:\text{Galois Extension}\quad(\because\text{Prop.ガロアの推進定理の補題(i)})$
$\text{Gal}(E/L) \simeq \text{Gal}(M/F')\quad(\because\text{Prop.ガロアの推進定理の補題(i)})$
$\therefore$
$E/F'$:Galois Extension
$E/F$:Galois Extension
$\text{Gal}(E/F') \simeq \text{Gal}(L/F') \times \text{Gal}(M/F')\quad(\because\text{Prop.ガロアの推進定理の補題(ii)})$