From 4ed629e651554b040fbd8328a9590e3c3e80df9b Mon Sep 17 00:00:00 2001 From: Gabriel Antonio Videtta Date: Thu, 14 Dec 2023 11:37:03 +0100 Subject: [PATCH] feat(algebra1/scheda): gruppi di Galois noti, teoremi fondamentali --- .../main.pdf | Bin 200663 -> 283125 bytes .../main.tex | 307 +++++++++++++++++- 2 files changed, 300 insertions(+), 7 deletions(-) diff --git a/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.pdf b/Secondo anno/Algebra 1/3. 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Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex index 4e4ac36..a86042f 100644 --- a/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex +++ b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex @@ -153,10 +153,17 @@ di polinomi di $K[x]$ un sovracampo minimale per inclusione di $K$ che fa sì che ogni polinomio di $\mathcal{F}$ si decomponga in fattori lineari. I campi di spezzamento di $\mathcal{F}$ sono sempre - $K$-isomorfi tra loro. Per il criterio della derivata, + $K$-isomorfi tra loro. \medskip + + + Un polinomio irriducibile si dice separabile se ammette + radici distinte. Per il criterio della derivata, $p \in K[x]$ ammette radici multiple se e solo se $\MCD(p, p')$ non è invertibile, dove $p'$ è la derivata - formale di $p$. \medskip + formale di $p$. Se $p \in K[x]$ e $n := \deg p$, il campo di spezzamento $L$ di $p$ è tale per cui + $[L : K] \leq n!$. Se $p$ è irriducibile e separabile, vale anche che $n \mid [L : K] \mid n!$, come + conseguenza dell'azione del relativo gruppo di + Galois sulle radici. \medskip Se $p$ è irriducibile in $K[x]$, $(p)$ è un ideale @@ -192,13 +199,41 @@ una $K$-immersione che è isomorfismo. \medskip + \subsection{Composto di estensioni e teorema delle torri algebriche} + + Date estensioni $L$ e $M$ su $K$, si definisce $LM = L(M) = M(L)$ come il \textbf{composto} di $L$ ed $M$, ossia come la più piccola estensione di $K$ che contiene sia $L$ che $M$. In particolare, $LM$ può essere visto come $L$-spazio vettoriale con vettori in $M$, o analogamente come $M$-spazio con - vettori in $L$. + vettori in $L$. \medskip + + + Per il Teorema delle torri algebriche, $L / K$ è + un'estensione finita se e solo se $L / F$ e + $F / K$ lo sono (ossia la finitezza vale strettamente + per torri). Inoltre, se $\basis_{L/F}$ e $\basis_{F/K}$ + sono basi di $L/F$ e $F/K$, allora + $\basis_{L/F} \basis_{F/K}$ è una base di + $L / K$, dove i suoi elementi sono i prodotti tra + i vari elementi delle due basi. Infine + se $L / K$ è finita, allora + anche $LM / M$ è finita, e vale che $[LM : M] \leq [L : K]$ (infatti una base di $L / K$ può essere trasformata + in un insieme di generatori di $LM / M$), e quindi + la finitezza vale per \textit{shift}. Sempre per + il Teorema delle torri algebriche, se $L / K$ è + finito, allora vale che: + \[ [L : K] = [L : F] [F : K]. \] + Se $L / K$ e $M / K$ sono finite, anche $LM / K$ lo + è (infatti la finitezza vale sia per torri che per \textit{shift}). In particolare, vale che: + \[ \mcm([L : K], [M : K]) \mid [LM : K]. \] + Se $[L : K]$ ed $[M : K]$ sono coprimi tra loro, + allora vale proprio l'uguaglianza + $[LM : K] = [L : K] [M : K]$. Infatti, in tal caso, + si avrebbe $[L : K] [M : K] \leq [LM : K]$ e + $[LM : K] = [LM : M] [M : K] \leq [L : K] [M : K]$. \subsection{Omomorfismo di valutazioni, elementi algebrici e trascendenti e polinomio minimo} @@ -257,7 +292,8 @@ $K$ se per ogni elemento $\alpha \in L$, $\mu_\alpha$ ammette radici distinte. Si dice che $K$ è un \textbf{campo perfetto} se ogni - polinomio irriducibile ammette radici distinte. + polinomio irriducibile ammette radici distinte, + ossia se ogni polinomio irriducibile è separabile. In un campo perfetto, ogni estensione algebrica è separabile. Si definiscono i coniugati di $\alpha$ algebrico su $K$ come le radici @@ -297,7 +333,7 @@ $\overline{\RR} = \CC$. - \subsection{Estensioni normali e $K$-immersioni di un'estensione finita di $K$} + \subsection{Estensioni normali e di Galois, $K$-immersioni di un'estensione finita di $K$} Sia $\alpha$ un elemento algebrico su $K$. Allora $[K(\alpha) : K] = \deg_K \alpha$. Le @@ -308,7 +344,7 @@ in $\overline{K}$. \medskip - Se $L / K$ è un'estensione finita su $K$, allora + Se $L / K$ è un'estensione separabile finita su $K$, allora esistono esattamente $[L : K]$ $K$-immersioni da $L$ in $\overline{K}$. Per quanto detto prima, tali immersioni mappano gli elementi $L$ nei @@ -321,6 +357,17 @@ tali per cui $\restr{\varphi_i}{K} = \varphi$. \medskip + Per quanto detto prima, per calcolare dunque tutti + i coniugati di $\alpha \in L$ su $K$, è sufficiente + calcolare i distinti valori delle $K$-immersioni + di $L$ su $\alpha$. Infatti, ogni $K$-immersione + da $K(\alpha)$ può estendersi a $K$-immersione di + $L$, e viceversa ogni $K$-immersione di $L$ può + restringersi a $K$-immersione di $K(\alpha)$. In + particolare, una volta computati tutti i coniugati, è semplice trovare il polinomio minimo di $\alpha$ + su $K$ (è sufficiente considerare il prodotto dei vari $x-\alpha_i$ dove gli $\alpha_i$ sono tutti i coniugati di $\alpha$). \medskip + + Si dice che un'estensione algebrica $L / K$ è un'\textbf{estensione normale} se per ogni $K$-immersione $\varphi$ da $L$ in $\overline{K}$ vale che $\varphi(L) = L$. Equivalentemente @@ -336,15 +383,261 @@ un automorfismo di $L$ che fissa $K$. \medskip + Un'estensione finita $L/K$ di grado $2$ è sempre normale, + ed in particolare può sempre scriversi come + $L = K(\sqrt{\Delta})$, dove $\Delta$ non è un quadrato + in $K$. + + Si indica con $\Aut_K(L) = \Aut(L / K)$ l'insieme degli automorfismi di $L$ che fissano $K$. Se $L$ è normale e separabile, si dice \textbf{estensione di Galois}, e si definisce - $\Gal(L / K) := (\Aut_K L, \circ)$, ossia come + il suo \textbf{gruppo di Galois} + $\Gal(L / K)$ come $(\Aut_K L, \circ)$, ossia come il gruppo $\Aut_K L$ con l'operazione di composizione. + \subsection{Azione di $\Gal(L / K)$ sulle radici di $L$ campo di spezzamento} + + Sia $p \in K[x]$ irriducibile e separabile. + Allora si definisce + il \textbf{gruppo di Galois di $p$} come il gruppo + di Galois $\Gal(L / K)$, dove $L$ è un campo di + spezzamento di $p$ su $K$. Se $\deg p = n$ e + $a_1$, ..., $a_n$ sono le radici di $p$, + $\Gal(L / K)$ agisce su $\{a_1, \ldots, a_n\}$ + mediante $\Xi$, in modo tale che: + \vskip -0.3in + \begin{equation*} + \begin{split} + \Xi : \Gal(&L / K) \to S(\{a_1, \ldots, a_n\}) \cong S_n, \\ + &\varphi_i \xmapsto{\Xi} [a_j \mapsto \varphi_i(a_j)]. + \end{split} + \end{equation*} + \vskip -0.2in + In particolare tale azione è transitiva (dunque $\Orb(a_i) = \{a_j\}_{j=1-n}$)e fedele. Poiché $\Xi$ è fedele, vale che + $\Gal(L / K) \mono S_n$. Se $\Gal(L / K)$ è abeliano + (e in tal caso si dice che $L$ è un'\textbf{estensione abeliana}), $\Xi$ è anche transitiva, e quindi + $\Gal(L / K)$ si identifica come un sottogruppo + abeliano transitivo di $S_n$, e in quanto tale deve + valere che $\abs{\Gal(L / K)} = n$. \medskip + + + + Dal momento che $\Xi$ è un'immersione, vale + che $\abs{\Gal(L / K)} \mid n!$. Dacché allora + $[K(a_1) : K] = n$, vale in particolare che: + \[ n \mid \abs{\Gal(L / K)} = [L : K] \mid n!. \] + + + \section{Diagrammi di campo e proprietà} + + + Si definisce \textbf{diagramma di campo} un + diagramma della seguente forma: + \[\begin{tikzcd} + & LM \\ + L & {} & M \\ + & {L \cap M} \\ + & K + \arrow[no head, from=4-2, to=3-2] + \arrow[no head, from=3-2, to=2-1] + \arrow[no head, from=2-1, to=1-2] + \arrow[no head, from=3-2, to=2-3] + \arrow[no head, from=2-3, to=1-2] + \end{tikzcd}\] + In particolare il precedente diagramma rappresenta + lo studio dell'estensione di $LM$ su $K$, e + rappresenta $L$, $M$ e $L \cap M$ come sottoestensioni + di $LM$. Un estremo superiore di una freccia è sempre, + per definizione, un'estensione dell'estremo inferiore + della stessa freccia. \medskip + + + Sia $\mathcal{P}$ una proprietà. Allora si + studia la proprietà $\mathcal{P}$ secondo + le seguenti tre modalità: + \begin{itemize} + \item validità per \textbf{torri}: se $\mathcal{P}$ vale in due estensioni in $K \subseteq F \subseteq L$, allora vale anche per la terza estensione, ossia + vale per tutta la torre di estensioni, + \item validità per \textbf{\textit{shift}} (o per il \textbf{traslato}): se $\mathcal{P}$ vale + per $F / K$, allora vale anche per $LF / F$, ossia + vale sul ramo parallelo a quello di $F / K$, + \item validità per il \textbf{composto}: se + $\mathcal{P}$ vale per $L / K$ ed $M / K$, allora + vale anche per $LM / K$. + \item validità per l'\textbf{intersezione}: + se $\mathcal{P}$ vale per $L / K$ ed $M / K$, + allora vale anche per $L \cap M / K$. + \end{itemize} + Si dice che $\mathcal{P}$ vale \textit{debolmente} + per torri, se $\mathcal{P}$ vale per $L / K$ solo + se vale per $L / F$ sottoestensione. + Si dice che $\mathcal{P}$ vale \textit{strettamente} + per torri, se è $\mathcal{P}$ vale per $L / K$ se + e solo se vale per $L / F$ e $F / K$. Se $\mathcal{P}$ vale strettamente per torri, allora $\mathcal{P}$ + vale anche per l'intersezione. \medskip + + + Si dice che + $\mathcal{P}$ vale \textit{inversamente} per + \textit{shift} se $\mathcal{P}$ vale su + $LF / F$ solo se vale su $L / K$. Si dice che + $\mathcal{P}$ vale \textit{inversamente} per + il composto se $\mathcal{P}$ vale su $LF / K$ + implica che $\mathcal{P}$ valga anche su $L / K$ + e $F / K$. Si dice che $\mathcal{P}$ vale \textit{completamente} per \textit{shift} o composto se $\mathcal{P}$ + vale \textit{inversamente} e normalmente per \textit{shift} o + composto. Se $\mathcal{P}$ vale per torri e + per \textit{shift}, allora vale anche per il + composto. + + La seguente tabella raccoglie le proprietà + delle estensioni sui diagrammi di campo: + \begin{center} + \scriptsize + \vskip -0.1in + \begin{tabular}{l|l|l|l|l} + \hline + $\mathcal{P}$ & Torri & \textit{Shift} & Composto & Intersez. \\ \hline + Est. fin. & Strett. & Normal. & Complet. & Sì \\ \hline + Est. alg. & Strett. & Complet. & Complet. & Sì \\ \hline + Est. sep. & Strett. & Normal. & Normal. & Sì \\ \hline + Est. nor. & Debolm. & Normal. & Normal. & Sì \\ \hline + Est. Gal. & Debolm. & Normal. & Normal. & Sì + \end{tabular} + \end{center} + + + \section{Teorema dell'elemento primitivo} + + Se $L / K$ è un'estensione finita e separabile, + $L$ è in particolare un'estensione semplice di + $K$, per il \textbf{Teorema dell'elemento primitivo}. + In campi finiti, un tale elemento primitivo è + un generatore di $L^*$. In campi infiniti, per + $L = K(a, b)$, + si può invece considerare il seguente polinomio: + \[ p(x) = \prod_{i < j} (\varphi_i(a) + x \varphi_i(b) - \varphi_j(b) - x \varphi_j(b)), \] + dove le varie $\varphi_i$ sono le $K$-immersioni di + $L$ su $\overline{K}$. + Si verifica che $p(x)$ è non nullo, e pertanto + ha supporto non vuoto. Pertanto esiste un $t \in K$ tale + per cui $p(t) \neq 0$, da cui si ricava che + $L = K(a + bt)$. Reiterando questo algoritmo su + tutti i generatori dell'estensione, si ottiene + un elemento primitivo desiderato. + + \section{Teorema di corrispondenza di Galois} + + + Se $L / K$ è di Galois, detto $H \leq \Gal(L / K)$, + si definisce $L^H$ come la sottoestensione di $L$ + fissata da tutte le $K$-immersioni di $H$. + In particolare vale che $L^H = K \iff H = \Gal(L / K)$. + Conseguentemente, vale il \textbf{Teorema di corrispondenza di Galois}, di seguito descritto: + + \begin{theorem} + Sia $\mathcal{E}$ l'insieme delle sottoestensioni + di $L / K$ estensione di Galois. Sia + $\mathcal{G}$ l'insieme dei sottogruppi di + $\Gal(L / K)$. Allora $\mathcal{E}$ è + in bigezione con $\mathcal{G}$ attraverso + la mappa $\alpha : \mathcal{E} \to \mathcal{G}$ + tale per cui: + \[ F \xmapsto{\alpha} \Gal(L / F) \leq + \Gal(L / K), \] + la cui inversa $\beta : \mathcal{G} \to \mathcal{E}$ + è tale per cui: + \[ H \xmapsto{\beta} L^H \subseteq L. \] + Inoltre, una sottoestensione $F / K$ di + $L / K$ è normale su $K$ se e solo se + il corrispondente sottogruppo di $\Gal(L / K)$ + è normale. Infine, se $F / K$ è normale, + $F$ è in particolare di Galois e vale che: + \[ \Gal(F / K) \cong \faktor{\Gal(L / K)}{\Gal(L / F)}. \] + \end{theorem} + + Pertanto, a partire dal Teorema di corrispondenza di Galois, valgono le seguenti proprietà: + + \begin{itemize} + \item il numero di sottogruppi di $\Gal(L / K)$ di un certo ordine $n$ è uguale al numero di sottoestensioni di $L$ tali per cui $L$ abbia + grado $n$ su di esse (infatti $[L : F] = \abs{\Gal(L / F)}$), + \item il numero di sottogruppi di $\Gal(L / K)$ di + un certo indice $n$ è uguale al numero di + sottoestensioni di $L$ che hanno grado $n$ su + $K$ (infatti $[F : K] = [L : K] / [L : F] = \abs{\Gal(L / K)}) / \abs{\Gal(L : F)} = + [\Gal(L / K) : \Gal(L / F)]$), + \item $L^H \subset L^Q \iff Q < H$, + \item $L^H L^Q = L^H(L^Q) = L^{H \cap Q}$, + \item $L^{\gen{H, Q}} = L^H \cap L^Q$, + \end{itemize} + + In particolare, un diagramma di campi -- a patto + che il suo estremo superiore sia di Galois -- può + essere collegato ad un diagramma di gruppi, + ``invertendo'' le inclusioni. Se + $G = \Gal(L / K)$ e $H \subseteq G$, allora il + diagramma: + \[\begin{tikzcd} + L \\ + {L^H} \\ + K + \arrow["G", bend left, no head, from=3-1, to=1-1] + \arrow["{G/H}"', no head, from=3-1, to=2-1] + \arrow["H"', no head, from=2-1, to=1-1] + \end{tikzcd}\] + si relaziona tramite corrispondenza al + diagramma: + \[\begin{tikzcd}[row sep=small] + {\{e\}} \\ + \\ + H \\ + \\ + G + \arrow[no head, from=1-1, to=3-1] + \arrow[no head, from=3-1, to=5-1] + \end{tikzcd}\] + + \section{Gruppi di Galois noti} + + \subsection{Campi finiti} + + Il campo finito $\FF_{p^n}$ è sempre normale + su $\FF_p$, dal momento che può essere costruito + come campo di spezzamento di $x^{p^n} - x$ su + $\FF_p$ stesso. Equivalentemente, poiché + un omomorfismo di campi è sempre iniettivo (e dunque + conserva sempre la cardinalità), + una $\FF_p$-immersione deve mandare $\FF_{p^n}$ + in un campo della stessa cardinalità, e quindi + necessariamente un campo isomorfo a $\FF_{p^n}$. \medskip + + + Per un campo finito, $\Frob$ è un automorfismo che + fissa $\FF_p$. Allora $\Frob \in \Gal(\FF_{p^n} / \FF_p)$. Inoltre $\ord \Frob = n = \abs{\Gal(\FF_{p^n} / \FF_p}$ (altrimenti $\FF_{p^n}$ non sarebbe campo di + spezzamento di $x^{p^n}-x$), e quindi vale che: + \[ \Gal(\FF_{p^n} / \FF_p) = \gen{\Frob} \cong \ZZmod{n}. \] + + + Pertanto se $\alpha \in \FF_{p^n} \setminus \FF_p$, + tutti i suoi coniugati si ottengono reiterando + al più $p^n$ volte $\Frob$ su $\alpha$. + + \subsection{Polinomi biquadratici} + + Sia $p(x) = x^4 + ax^2 + b$ irriducibile su $\QQ$. + Allora, se $L$ è un suo campo di spezzamento e $\Delta = a^2 - 4b$ è l'usuale discriminante di $p$ visto come polinomio in $x^2$, vale che: + + \[ \Gal(L / \QQ) \cong \begin{cases} + \ZZmod{4} & \se b \text{ è quadrato in $\QQ$}, \\ + \ZZmod{2} \times \ZZmod{2} & \se b \Delta \text{ è quadrato in $\QQ$}, \\ + D_4 & \altrimenti. + \end{cases} \] + + \vfill \hrule ~\\