From add484dfa6293dde1260af59663e57111965fd32 Mon Sep 17 00:00:00 2001 From: Gabriel Antonio Videtta Date: Thu, 7 Dec 2023 12:36:35 +0100 Subject: [PATCH] feat(algebra1): teorema di corrispondenza di Galois --- .../main.pdf | Bin 185870 -> 173159 bytes .../2. Il gruppo delle permutazioni/main.pdf | Bin 266398 -> 254167 bytes .../2. Il gruppo delle permutazioni/main.tex | 10 + .../main.pdf | Bin 168679 -> 227646 bytes .../main.tex | 238 +++++++++++++++++- 5 files changed, 247 insertions(+), 1 deletion(-) diff --git a/Secondo anno/Algebra 1/1. Teoria dei gruppi/Azioni di gruppo e p-gruppi/1. Azione di un gruppo su un insieme/main.pdf b/Secondo anno/Algebra 1/1. Teoria dei gruppi/Azioni di gruppo e p-gruppi/1. 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Teoria dei gruppi/Gruppi classici (Aut(G), Sn, Dn, G_ab, Q8)/2. Il gruppo delle permutazioni/main.tex b/Secondo anno/Algebra 1/1. Teoria dei gruppi/Gruppi classici (Aut(G), Sn, Dn, G_ab, Q8)/2. Il gruppo delle permutazioni/main.tex index ecaad56..941fa1d 100644 --- a/Secondo anno/Algebra 1/1. Teoria dei gruppi/Gruppi classici (Aut(G), Sn, Dn, G_ab, Q8)/2. Il gruppo delle permutazioni/main.tex +++ b/Secondo anno/Algebra 1/1. Teoria dei gruppi/Gruppi classici (Aut(G), Sn, Dn, G_ab, Q8)/2. Il gruppo delle permutazioni/main.tex @@ -43,6 +43,16 @@ a $\cap_{x=1}^n \Stab(x)$, e quindi se e solo se $\sigma = e$. Si conclude dunque che $\Stab(i)$ è banale e quindi che $\abs{H} = n$. \end{proof} + + \begin{proof}[Dimostrazione alternativa] + Se $H$ è un sottogruppo transitivo di $S_n$, + allora la sua azione naturale agisce fedelmente e + transitivamente su $X_n$. Poiché però $H$ è anche + abeliano, l'azione è anche libera, e dunque ogni + stabilizzatore è banale. Pertanto, per il + Teorema orbita-stabilizzatore, $\abs{H} = + \abs{\Stab(1)} \abs{\Orb(1)} = n$. + \end{proof} \begin{example}[Il gruppo di Klein $V_4$] In $S_4$, e in particolare in $A_4$, esiste un sottogruppo normale non banale diff --git a/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.pdf b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. 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KdI?2I?EeSi2Ewuc diff --git a/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.tex b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.tex index b60dfb6..1cc1730 100644 --- a/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.tex +++ b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.tex @@ -92,7 +92,243 @@ $[K(\gamma) : K] = n = [K(\alpha, \beta) : K]$, da cui $K(\alpha, \beta) = K(\gamma)$, ossia la tesi. \end{itemize} + \end{proof} \medskip + + + Si illustrano adesso i prerequisiti per dimostrare + il Teorema di corrispondenza di Galois: + + \begin{definition} + Sia $\faktor{L}{K}$ un'estensione di Galois. Allora, + se $H \leq \Gal(\faktor{L}{K})$, si definisce + $L^H = \Fix(H)$ come la sottoestensione di $L$ su + $K$ degli elementi fissati da ogni $\varphi \in H$, + ossia: + \[ L^H = \{ \alpha \in L \mid \varphi(\alpha) = \alpha \, \forall \varphi \in H \}. \] + \end{definition} + + \begin{nlemma} + Sia $\faktor{L}{K}$ un'estensione di Galois. Allora, + se $H \leq \Gal(\faktor{L}{K})$ vale che: + \[ L^H = K \iff H = \Gal(\faktor{L}{K}). \] + \end{nlemma} + + \begin{proof} + Sia $H = \Gal(\faktor{L}{K})$. Allora sicuramente + $K \subseteq L^H$. Si mostra che non può valere + $K \subsetneq L^H$. Se infatti $K \subsetneq L^H$, + varrebbe che $[L^H : K] > 1$, e quindi esisterebbe + una $K$-immersione non banale di $L^H$, detta + $\varphi : L^H \to \overline{K}$. In particolare + $\varphi$ può estendersi a una $K$-immersione di + $L$, detta $\tilde{\varphi}$. In particolare + $\tilde{\varphi} \in \Gal(\faktor{L}{K})$, e + quindi $\tilde{\varphi}$ deve fissare $L^H$ per + ipotesi. Tuttavia $\tilde{\varphi}$ ristretta + a $L^H$ non fissa $L^H$ per ipotesi, \Lightning. + Pertanto $L^H = K$. \medskip + + + Sia adesso $L^H = K$. Per il Teorema dell'elemento + primitivo, $\exists \alpha \in L^H$ tale per cui + $L = K(\alpha)$. Si consideri allora il + polinomio $p$ a coefficienti in $\overline{K}$ tale + per cui: + \[ p(x) = \prod_{\varphi \in H} (x - \varphi(\alpha)). \] + Poiché l'identità di $\Gal(\faktor{L}{K})$ appartiene + ad $H$, $(x-\alpha) \mid p(x)$, e quindi + $p(\alpha) = 0$. Inoltre $p$ è in realtà un polinomio + a coefficienti in $L^H$. Se infatti $\rho \in H$, + \[ \rho(p(x)) = \prod_{\varphi \in H} (x - \rho(\varphi(\alpha))) = p(x), \] + dove l'uguaglianza è dovuta al fatto\footnote{ + In particolare è stato applicato l'\textit{embedding} di Cayley su $H$ + attraverso l'elemento $\rho \in H$, e + quest'azione si è rivelata essere transitiva. + } che le + mappe $\{ \rho \circ \varphi \}$ sono esattamente le + mappe $\{ \varphi \}$. Pertanto $\abs{\Gal(\faktor{L}{K})} = [L : K] = + [K(\alpha) : K] \leq \deg p(x) = \abs{H}$ dal + momento che $\alpha$ è radice di $p(x)$. Dal momento + che vale anche che $\abs{\Gal(\faktor{L}{K})} \geq \abs{H}$, allora + $H = \Gal(\faktor{L}{K})$, da cui la tesi. + \end{proof} + + \begin{proposition} + Sia $\sigma \in \Gal{\faktor{L}{K}}$. Allora, + se $H \leq \faktor{L}{K}$, vale che + $\sigma(L^H) = L^{\sigma H \sigma\inv}$. + \end{proposition} + + \begin{proof} + Si osserva che: + \[ \sigma(L^H) = \{ \sigma(\alpha) \mid \alpha \in L, \, \varphi(\alpha) = \alpha \, \forall \varphi \in H \} = \{ \beta \in L \mid \varphi(\sigma\inv(\beta)) = \sigma\inv(\beta) \, \forall \varphi \in H \}, \] + dove si è sfruttato in modo cruciale il fatto che $\varphi \in H$ è bigettiva. Si + conclude allora che: + \[ \varphi(L^H) = \{ \beta \in L \mid \sigma(\varphi(\sigma\inv(\beta))) = \beta \, \forall \varphi \in H \} = L^{\sigma H \sigma\inv}. \] \end{proof} + + Si può adesso dimostrare il Teorema di corrispondenza di + Galois: - %TODO: aggiungere corrispondenza di Galois + \begin{theorem}[di corrispondenza di Galois] + Sia $\mathcal{E}$ l'insieme delle sottoestensioni + di $\faktor{L}{K}$ estensione di Galois. Sia + $\mathcal{G}$ l'insieme dei sottogruppi di + $\Gal(\faktor{L}{K})$. Allora $\mathcal{E}$ è + in bigezione con $\mathcal{G}$ attraverso + la mappa $\alpha : \mathcal{E} \to \mathcal{G}$ + tale per cui: + \[ F \xmapsto{\alpha} \Gal(\faktor{L}{F}) \leq + \Gal(\faktor{L}{K}), \] + la cui inversa $\beta : \mathcal{G} \to \mathcal{E}$ + è tale per cui: + \[ H \xmapsto{\beta} L^H \subseteq L. \] + Inoltre, una sottoestensione $\faktor{F}{K}$ di + $\faktor{L}{K}$ è normale su $K$ se e solo se + il corrispondente sottogruppo di $\Gal(\faktor{L}{K})$ + è normale. Infine, se $\faktor{F}{K}$ è normale, + $F$ è in particolare di Galois\footnote{ + Si ricorda che si considera $K$ un campo perfetto. + } e vale che: + \[ \Gal(\faktor{F}{K}) \cong \faktor{\Gal(\faktor{L}{K})}{\Gal(\faktor{L}{F})}. \] + \end{theorem} + + \begin{proof} + Le mappe $\alpha$ e $\beta$ sono ovviamente ben definite. Si mostra direttamente che sono l'una + l'inversa dell'altra. Sia $H \leq \Gal(\faktor{L}{K})$. + Si osserva che: + \[ \alpha(\beta(H)) = \alpha(L^H) = \Gal(\faktor{L}{L^H}). \] + Sia $L^H = M$. Se si pone $K = \Gal(\faktor{L}{L^H})$, vale chiaramente che $H \leq K$ dal momento che $H$ + fissa per definizione tutti gli elementi di + $L^H$. Dacché allora $L^H = M$, per il lemma precedente + $H = K$, e quindi $\alpha(\beta(H)) = H$. \medskip + + + Analogamente si osserva che per $K \subseteq F \subseteq K$ vale che: + \[ \beta(\alpha(F)) = \beta(\Gal(\faktor{L}{F})) = + L^{\Gal(\faktor{L}{F})}. \] + Pertanto, detto $H = \Gal(\faktor{L}{F})$, + per il lemma precedente vale che $L^H = F$, + e quindi $\beta(\alpha(F)) = F$, dimostrando + la prima parte del teorema. \medskip + + + Sia ora $\faktor{F}{K}$ una sottoestensione normale + di $\faktor{L}{K}$. Allora, se $\varphi \in \Gal(\faktor{L}{F})$ e $\sigma \in \Gal(\faktor{L}{K})$, + $\tau = \sigma \circ \varphi \circ \sigma\inv$ è ancora + un elemento di $\faktor{L}{K}$. Pertanto, + $\tau$ si può restringere ad una $K$-immersione di $F$. + Poiché allora $F$ è normale su $K$, $\tau(F) = F$, e + quindi $\tau \in \Gal(\faktor{L}{F})$, e dunque + $\Gal(\faktor{L}{F}) \nsgeq \Gal(\faktor{L}{K})$. \medskip + + + Sia adesso $\Gal(\faktor{L}{F}) \nsgeq \Gal(\faktor{L}{K})$. Sia $\varphi$ una $K$-immersione + di $F$ su $\overline{K}$. Allora $\varphi$ può essere + estesa ad un elemento $\tilde{\varphi} \in \Gal(\faktor{L}{K})$. In particolare, + se $H = \Gal(\faktor{L}{F})$, $\varphi(F) = + \tilde{\varphi}(F) = L^{\varphi H \varphi\inv} = L^H = F$, dove si è sfruttata la normalità di $H$ in + $\Gal(\faktor{L}{K})$. Pertanto $F$ è normale su $K$, + e dunque, in quanto separabile per ipotesi, di Galois. \medskip + + + Si consideri adesso l'omomorfismo $\tau : \Gal(\faktor{L}{K}) \to \Gal(\faktor{F}{K})$ dato + dalla restrizione delle immersioni di $\Gal(\faktor{L}{K})$ su $F$. Chiaramente $\tau$ + è una mappa surgettiva, dal momento che ogni $K$-immersione di $\Gal(\faktor{F}{K})$ può estendersi + a $K$-immersione di $\Gal(\faktor{L}{K})$. Inoltre + vale che $\Ker \tau$ è esattamente il sottogruppo + di $\Gal(\faktor{L}{K})$ che fissa $F$, ossia + $\Gal(\faktor{L}{F})$. Applicando allora il Primo + teorema di isomorfismo vale che: + \[ \Gal(\faktor{F}{K}) \cong \faktor{\Gal(\faktor{L}{K})}{\Gal(\faktor{L}{F})}, \] + da cui la tesi. + \end{proof} + + \begin{example}[studio dei sottocampi di $\QQ(\sqrt{2}, \sqrt{3}) \quot \QQ$] + Dal momento che $L := \QQ(\sqrt{2}, \sqrt{3})$ è + il campo di spezzamento dei polinomi $x^2 - 2$ + e $x^2 - 3$, tale estensione è normale su $\QQ$, + e quindi di Galois. Inoltre, dal momento che + $\sqrt{3} \notin \QQ(\sqrt{2})$, + $[L : \QQ] = 2 \cdot 2 = 4$, dal Teorema delle + torri algebriche. Pertanto $\Gal(\faktor{L}{\QQ})$ + è un gruppo di ordine $4$. \medskip + + + Si definisce $\varphi_{ij}$ con $i$, $j \in \{0, 1\}$ + come le $\QQ$-immersioni di $L$ + tali per cui $\sqrt{2} \xmapsto{\varphi_{ij}} (-1)^i \sqrt{2}$ e analogamente $\sqrt{3} \xmapsto{\varphi_{ij}} (-1)^j \sqrt{3}$. Dal momento + che le varie $\varphi_{ij}$ sono distinte, + che ogni $\varphi_{ij}$ ha ordine $2$ e che ogni + gruppo di ordine $4$ è abeliano (o, più semplicemente, + le varie $\varphi_{ij}$ commutano tra loro), + vale che $\Gal(\faktor{L}{\QQ}) \cong \ZZmod2 \times \ZZmod2$. \medskip + + + Ogni sottoestensione di $L$ ha + grado su $\QQ$ divisore di $[L : \QQ]$, e quindi + ha grado $1$, $2$ o $4$. Se il grado è $4$, + la sottoestensione considerata è proprio $L$, mentre + se il grado è $1$ la sottoestensione è $\QQ$ stesso. + Si studiano ora le sottoestensioni di grado $2$. + Tali sottoestensioni corrispondono ai sottogruppi + di $\Gal(\faktor{L}{\QQ})$ di ordine $4/2 = 2$. Inoltre, + a priori, essendo $\Gal(\faktor{L}{\QQ})$ abeliano, tutte + le sottoestensioni sono normali su $\QQ$. \medskip + + + Ogni sottogruppo di ordine $2$ è ciclico e generato + da elementi di ordine $2$, e quindi, mantenendo + la corrispondenza con $\ZZmod2 \times \ZZmod2$, + da $(1,0)$, $(0,1)$ o $(1,1)$. Pertanto esistono + esattamente $3$ sottoestensioni distinte di grado $2$ + su $\QQ$. \medskip + + + In particolare queste sottoestensioni corrispondono + ai sottocampi di $L$ fissati da $\varphi_{10}$, + $\varphi_{01}$ e $\varphi_{11}$, ossia + $\QQ(\sqrt{3})$, $\QQ(\sqrt{2})$ e $\QQ(\sqrt{6})$. \medskip + + + Inoltre $\alpha := \sqrt{2} + \sqrt{3}$ è un elemento primitivo + di $L$, dal momento che non può appartenere né a + $\QQ(\sqrt{3})$ né a $\QQ(\sqrt{2})$ (altrimenti + tali sottoestensioni coinciderebbero con $L$, \Lightning), e così nemmeno a $\QQ(\sqrt{6})$ (altrimenti $\alpha$ si scriverebbe + come combinazione lineare di $1$ e $\sqrt{6}$, + \Lightning). Alternativamente $\alpha$ + ha esattamente $4$ coniugati tramite + le varie\footnote{ + Tali $4$ coniugati sono distinti dal momento + che $\{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\}$ è + una base di $L$ come $\QQ$-spazio. + } $\varphi_{ij}$, e quindi ha grado $4$ su + $\QQ$. In particolare vale che: + \[ \mu_\alpha(x) = \prod_{i=0}^1 \prod_{j=0}^1 (x + (-1)^i \sqrt{2} + (-1)^j \sqrt{3}) = x^4 - 10x^2 + 1. \] + + Tutte le informazioni sono infine raccolte nel seguente + diagramma di estensioni: %TODO: terminare esempio + + + \[\begin{tikzcd}[column sep=2.25em] + && {\overbrace{\mathbb{Q}(\sqrt{2}, \sqrt{3})}^{\mathbb{Q}(\sqrt{2} + \sqrt3)}} \\ + \\ + {\mathbb{Q}(\sqrt{2})} && {\mathbb{Q}(\sqrt6)} && {\mathbb{Q}(\sqrt3)} \\ + \\ + && {\mathbb{Q}} + \arrow["2"{pos=0.3}, no head, from=1-3, to=3-1] + \arrow["2"', no head, from=1-3, to=3-3] + \arrow["2"', no head, from=1-3, to=3-5] + \arrow["2"', no head, from=3-3, to=5-3] + \arrow["2"{description}, no head, from=3-5, to=5-3] + \arrow["2"{description}, no head, from=3-1, to=5-3] + \arrow["{\small x^4 - 10x^2 + 1}", curve={height=-24pt}, no head, from=1-3, to=5-3] + \arrow["{x^2-3}", curve={height=-12pt}, no head, from=3-5, to=5-3] + \arrow["{x^2-2}"', curve={height=12pt}, no head, from=3-1, to=5-3] + \arrow["{x^2-6}"'{pos=0.3}, shift left, curve={height=12pt}, no head, from=3-3, to=5-3] + \arrow["{x^2+2\sqrt3\,x+1}", curve={height=-18pt}, no head, from=1-3, to=3-5] + \arrow["{x^2+2\sqrt2\,x-1}"', curve={height=18pt}, no head, from=1-3, to=3-1] + \arrow["{\small x^2-2\sqrt6-5}"'{pos=0.8}, curve={height=12pt}, no head, from=1-3, to=3-3] + \end{tikzcd}\] + \end{example} \end{document} \ No newline at end of file