From f3dfd19baa97de32970e089b7dba10639029ef52 Mon Sep 17 00:00:00 2001 From: hearot Date: Fri, 27 Mar 2026 19:27:08 +0100 Subject: [PATCH] alg1(galois): aggiorna la scheda riassuntiva --- .../main.pdf | Bin 314256 -> 285555 bytes .../main.tex | 1218 +++++++++-------- .../notes_2023.sty | 451 ++++++ 3 files changed, 1061 insertions(+), 608 deletions(-) create mode 100644 Corsi/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/notes_2023.sty diff --git a/Corsi/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.pdf b/Corsi/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/Corsi/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex index 14f15c1..1c79431 100644 --- a/Corsi/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex +++ b/Corsi/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex @@ -39,559 +39,524 @@ \title{Scheda riassuntiva di Teoria dei campi e di Galois} \begin{document} - - \parskip=0.7ex - - \raggedright - \footnotesize - - \begin{center} - \Large{\textbf{Scheda riassuntiva di Teoria dei campi e di Galois}} \\ - \end{center} - \begin{multicols}{3} - \setlength{\premulticols}{1pt} - \setlength{\postmulticols}{1pt} - \setlength{\multicolsep}{1pt} - \setlength{\columnsep}{2pt} - - \section{Campi e omomorfismi} - - Si dice \textbf{campo} un anello commutativo non banale - $K$ che è - contemporaneamente anche un corpo. Si dice - \textbf{omomorfismo di campo} tra due campi $K$ ed $L$ - un omomorfismo di anelli. Dal momento che un omomorfismo - $\varphi$ è tale per cui $\Ker \varphi$ è un ideale - di $K$ con $1 \notin \Ker \varphi$, deve per forza - valere $\Ker \varphi = \{0\}$, e quindi ogni omomorfismo - di campi è un'immersione. \medskip - - - \section{Caratteristica di un campo} - - Dato l'omomorfismo $\zeta : \ZZ \to K$ completamente - determinato dalla relazione $1 \xmapsto{\zeta} 1_K$, - si definisce \textbf{caratteristica di $K$}, detta - $\Char K$, il - generatore non negativo di $\Ker \zeta$. In particolare - $\Char K$ è $0$ o un numero primo. Se $\Char K$ è zero, - $\zeta$ è un'immersione, e quindi $K$ è un campo infinito, - e in particolare vi si immerge anche $\QQ$. \medskip - - - Tuttavia non è detto che $\Char K = p$ implichi che $K$ è - finito. In particolare $\ZZ_p(x)$, il campo delle funzioni - razionali a coefficienti in $\ZZ_p$, è un campo infinito - a caratteristica $p$. - - - \subsection{Proprietà dei campi a caratteristica $p$} - - Se $\Char K = p$, per il Primo - teorema di isomorfismo per anelli, $\ZZmod{p}$ si immerge - su $K$ tramite la proiezione di $\zeta$; pertanto - $K$ contiene una copia isomorfa di $\ZZmod{p}$. Per - campi di caratteristica $p$, vale il Teorema del - binomio ingenuo, ossia: - \[ (a + b)^p = a^p + b^p, \] - estendibile anche a più addendi. - In particolare, per un campo $K$ di caratteristica $p$, - la mappa $\Frob : K \to K$ tale per cui $a \xmapsto{\Frob} a^p$ - è un omomorfismo di campi, ed in particolare è un'immersione - di $K$ in $K$, detta \textbf{endomorfismo di Frobenius}. Se $K$ è un campo finito, $\Frob$ è anche - un isomorfismo. Si osserva che per gli elementi della - copia $K \supseteq \FF_p \cong \ZZmod{p}$ vale - $\restr{\Frob}{\FF_p} = \Id_{\FF_p}$, e quindi - $\Frob$ è un elemento di $\Gal(K / \FF_p)$. - - - \section{Campi finiti} - - - Per ogni $p$ primo e $n \in \NN^+$ esiste un campo finito - di ordine $p^n$. In particolare, tutti i campi finiti di - ordine $p^n$ sono isomorfi tra loro, possono essere visti - come spazi vettoriali di dimensione $n$ sull'immersione di $\ZZmod{p}$ che contengono, - e come campi di spezzamento di $x^{p^n}-x$ - su tale immersione. Tali campi hanno obbligatoriamente - caratteristica $p$, dove $\abs{K} = p^n$. Esiste - sempre un isomorfismo tra due campi finiti che manda la copia isomorfa di $\ZZmod{p}$ di uno nell'altra. \medskip - - - Poiché i campi finiti di medesima cardinalità sono isomorfi, - si indicano con $\FF_p$ e $\FF_{p^n}$ le strutture - algebriche di tali campi. In particolare con - $\FF_{p^n} \subseteq \FF_{p^m}$ si intende che - esiste un'immersione di un campo con $p^n$ elementi in - uno con $p^m$ elementi, e analogamente si farà con - altre relazioni (come l'estensione di campi) - tenendo bene in mente di star - considerando tutti i campi di tale ordine. \medskip - - - Vale la relazione $\FF_{p^n} \subseteq \FF_{q^m}$ - se e solo se $p=q$ e $n \mid m$. Conseguentemente, - l'estensione minimale per inclusione comune a - $\FF_{p^{n_1}}$, ..., $\FF_{p^{n_i}}$ è - $\FF_{p^m}$ dove $m := \mcm(n_1, \ldots, n_i)$. Pertanto - se $p \in \FF_{p^n}[x]$ si decompone in fattori irriducibili - di grado $n_1$, \ldots, $n_i$, il suo campo di spezzamento - è $\FF_{p^m}$. Inoltre, $x^{p^n}-x$ è in $\FF_p$ il - prodotto di tutti gli irriducibili di grado divisore - di $n$. - - \section{Proprietà dei polinomi di $K[x]$} - - Per il Teorema di Lagrange sui campi, ogni polinomio - di $K[x]$ ammette al più tante radici quante il suo grado. - Come conseguenza pratica di questo teorema, ogni sottogruppo - moltiplicativo finito di $K$ è ciclico. Pertanto - $\FF_{p^n}^* = \gen{\alpha}$ per $\alpha \in \FF_{p^n}$, - e quindi $\FF_{p^n} = \FF_p(\alpha)$, ossia - $\FF_{p^n}$ è sempre un'estensione semplice su $\FF_p$. Si dice - \textbf{campo di spezzamento} di una famiglia $\mathcal{F}$ - 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. \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 $p$. 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 - massimale, e $K[x] / (p)$ è un campo che - ne contiene una radice, ossia $[x]$. In - particolare $K$ si immerge in $K[x] / (p)$, - e quindi tale campo può essere identificato come - un'estensione di $K$ che aggiunge una radice di $p$. - Se $K$ è finito, detta $\alpha$ la radice aggiunta - all'estensione, $L := K[x] / (p) \cong K(\alpha)$ contiene - tutte le radici di $p$ (ed è dunque il suo campo - di spezzamento). Infatti detto $[L : \FF_p] = n$, - $[x]$ annulla $x^{p^n}-x$ per il Teorema di Lagrange - sui gruppi, e quindi $p$ deve dividere $x^{p^n}-x$; - in tal modo $p$ deve spezzarsi in fattori lineari, - e quindi ogni radice deve già appartenere ad $L$. - In particolare, ogni estensione finita e semplice - di un campo finito è normale, e quindi di Galois. \medskip - - \section{Estensioni di campo} - - - Si dice che $L$ è un'estensione di $K$, e si indica - con $L / K$, se $L$ è un sovracampo di $K$, - ossia se $K \subseteq L$. Si indica con $[L : K] = \dim_K L$ la - dimensione di $L$ come $K$-spazio vettoriale. Si - dice che $L$ è un'estensione finita di $K$ se $[L : K]$ - è finito, e infinita altrimenti. Un'\textbf{estensione finita} - di un campo finito è ancora un campo finito. Un'estensione - è finita se e solo se è finitamente generata da elementi algebrici. Una $K$-immersione è un omomorfismo di campi - iniettivo da un'estensione di $K$ in un'altra estensione di $K$ che - agisce come l'identità su $K$. Un $K$-isomorfismo è - 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 $L$. \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} - - - Dato $\alpha$, si definisce $K(\alpha)$ il più piccolo - sovracampo di $K$ che contiene $\alpha$. Si definisce l'\textbf{omomorfismo di + +\parskip=0.7ex + +\raggedright +\footnotesize + +\begin{center} + \Large{\textbf{Scheda riassuntiva di Teoria dei campi e di Galois}} \\ +\end{center} +\begin{multicols}{3} + \setlength{\premulticols}{1pt} + \setlength{\postmulticols}{1pt} + \setlength{\multicolsep}{1pt} + \setlength{\columnsep}{2pt} + + \textit{La teoria si concentra sulle estensioni finite su campi perfetti, + benché si sia dato spazio anche a considerazioni su estensioni infinite + o non separabili.} + + \section{Campi e omomorfismi} + + Si dice \textbf{campo} un anello commutativo non banale + $K$ i cui elementi non nulli ammettono un inverso + moltiplicativo. Si dice + \textbf{omomorfismo di campo} tra due campi $K$ ed $L$ + un omomorfismo che è anche omomorfismo di anelli. \smallskip + + Dal momento che un omomorfismo + $\varphi$ è tale per cui $\Ker \varphi$ è un ideale + di $K$ con $1 \notin \Ker \varphi$, deve per forza + valere $\Ker \varphi = \{0\}$, e quindi ogni omomorfismo + di campi è sempre iniettivo. + + \section{Caratteristica di un campo} + + Esiste un unico omomorfismo $\ZZ \to K$ completamente + determinato dalla relazione $1 \mapsto 1_K$, e + si definisce \textbf{caratteristica di $K$}, detta + $\Char K$, il + generatore non negativo del nucleo di questa mappa. In particolare + $\Char K$ è $0$ o un numero primo. \smallskip + + Se $\Char K$ è zero, + $\zeta$ è un'immersione, e quindi $K$ è un campo infinito, + e in particolare vi si immerge anche $\QQ$. \smallskip + + Tuttavia non è detto che $\Char K = p$ implichi che $K$ è + finito. In particolare $\FF_p(x)$, il campo delle funzioni + razionali a coefficienti in $\FF_p$, è un campo infinito + a caratteristica $p$. + + \subsection{Proprietà dei campi a caratteristica \texorpdfstring{$p$}{p}} + + Se $\Char K = p$, per il Primo + teorema di isomorfismo per anelli, $\FF_p$ si immerge + su $K$ tramite il passaggio al quoziente di $\ZZ \to K$; pertanto + $K$ contiene una copia isomorfa di $\FF_p$. Per + campi di caratteristica $p$, vale il Teorema del + binomio ingenuo, ossia: + \[ (a + b)^p = a^p + b^p, \] + estendibile anche a più addendi. \smallskip + + In particolare, per un campo $K$ di caratteristica $p$, + la mappa $\Frob : K \to K$ tale per cui $a \xmapsto{\Frob} a^p$ + è un omomorfismo di campi, ed in particolare è un'immersione + di $K$ in $K$, detta \textbf{endomorfismo di Frobenius}. \smallskip + + Se $K$ è un campo finito, $\Frob$ è anche + un isomorfismo (gli omomorfismi tra campi sono iniettivi!). + Si osserva che per gli elementi di $\FF_p \subseteq K$ vale + $\restr{\Frob}{\FF_p} = \Id_{\FF_p}$, e quindi + $\Frob$ è un elemento di $\Gal(K / \FF_p)$. + + \section{Campi finiti} + + Per ogni $p$ primo e $n \in \NN^+$ esiste un campo finito + di ordine $p^n$. Ogni tale campo può essere visto + come spazi vettoriale di dimensione $n$ sull'immersione di $\FF_p$ che contiene, + e come campo di spezzamento di $x^{p^n}-x$ + su tale immersione. Tale campi ha obbligatoriamente + caratteristica $p$. \smallskip + + Fissati $p$ ed $n$, esiste un unico campo finito di cardinalità + $p^n$: esiste sempre un isomorfismo tra due campi finiti che + fissa le copie di $\FF_p$. Indichiamo con $\FF_{p^n}$ la struttura + algebrica di un campo di cardinalità $p^n$. \smallskip + + Vale la relazione $\FF_{p^n} \subseteq \FF_{q^m}$ + se e solo se $p=q$ e $n \mid m$ (segue facilmente dal Teorema + delle torri algebriche). Conseguentemente, + l'estensione minimale per inclusione comune a + $\FF_{p^{n_1}}$, ..., $\FF_{p^{n_i}}$ è + $\FF_{p^m}$ dove $m := \mcm(n_1, \ldots, n_i)$. Pertanto + se $p \in \FF_{p^n}[x]$ si decompone in fattori irriducibili + di grado $n_1$, \ldots, $n_i$, il suo campo di spezzamento + è $\FF_{p^m}$. Da ciò segue anche che $x^{p^n}-x$ è in $\FF_p$ il + prodotto di tutti gli irriducibili di grado divisore + di $n$. + + \section{Campi di spezzamento per polinomi su \texorpdfstring{$K$}{K}} + + Per il Teorema di Lagrange sui campi, ogni polinomio + di $K[x]$ ammette al più tante radici quante il suo grado. + Come conseguenza pratica di questo teorema, ogni sottogruppo + moltiplicativo finito di $K$ è ciclico. Pertanto + $\FF_{p^n}^* = \gen{\alpha}$ per $\alpha \in \FF_{p^n}$, + e quindi $\FF_{p^n} = \FF_p(\alpha)$, ossia + $\FF_{p^n}$ è sempre un'estensione semplice su $\FF_p$. \smallskip + + Si dice + \textbf{campo di spezzamento} di una famiglia $\mathcal{F}$ + 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 (ossia ammettono isomorfismi che fissano + il campo base $K$). \smallskip + + 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 $p$. \smallskip + + Se $p \in K[x]$ e $n := \deg p$, il campo di spezzamento $L$ di $p$ è tale per cui + $[L : K] \leq n!$ (infatti, presa una radice, la sua aggiunta al campo base + restituisce un'estensione di grado al più + $n$; dobbiamo poi considerare il polinomio rimanente di grado $n-1$ che aumenta l'estensione + di al più un fattore $n-1$; reiterando si ottiene la tesi). 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. \smallskip + + Se $p$ è irriducibile in $K[x]$, $(p)$ è un ideale + massimale, e $K[x] / (p)$ è un campo che + ne contiene una radice, ossia $\overline x$. In + particolare $K$ si immerge in $K[x] / (p)$, + e quindi tale campo può essere identificato come + un'estensione di $K$ che aggiunge una radice di $p$. + Se $K$ è finito, detta $\alpha$ la radice aggiunta + all'estensione, $L := K[x] / (p) \cong K(\alpha)$ contiene + tutte le radici di $p$ (ed è dunque il suo campo + di spezzamento). Infatti detto $[L : \FF_p] = n$, + $\overline x$ annulla $x^{p^n}-x$ per il Teorema di Lagrange + sui gruppi, e quindi $p$ deve dividere $x^{p^n}-x$; + in tal modo $p$ deve spezzarsi in fattori lineari, + e quindi ogni radice deve già appartenere ad $L$ + (alternativamente, esiste un unico campo di cardinalità $p^n$ a + meno di isomorfismo, e quindi necessariamente le radici devono + già appartenere in $\FF_{p^n}$). + In particolare, ogni estensione finita e semplice + di un campo finito è normale, e quindi di Galois. + + \section{Estensioni di campo} + + Si dice che $L$ è un'estensione di $K$, e si indica + con $L / K$, se $L$ è un sovracampo di $K$, + ossia se $K \subseteq L$. Si indica con $[L : K] = \dim_K L$ la + dimensione di $L$ come $K$-spazio vettoriale. Si + dice che $L$ è un'estensione finita di $K$ se $[L : K]$ + è finito, e infinita altrimenti. Un'\textbf{estensione finita} + di un campo finito è ancora un campo finito. Un'estensione + è finita se e solo se è finitamente generata da elementi algebrici. \smallskip + + Una $K$-immersione è un omomorfismo di campi + da un'estensione di $K$ in un'altra estensione di $K$ che + agisce come l'identità su $K$. Un $K$-isomorfismo è + una $K$-immersione che è isomorfismo. + + \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 $L$. \smallskip + + 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} + + Dato $\alpha$, si definisce $K(\alpha)$ il più piccolo + sovracampo di $K$ che contiene $\alpha$. Si definisce l'\textbf{omomorfismo di valutazione} $\varphi_{\alpha, K} : K[x] \to K[\alpha]$, detto - $\varphi_\alpha$ se $K$ è noto, l'omomorfismo completamente - determinato dalla relazione $p \xmapsto{\varphi_\alpha} p(\alpha)$. Si verifica che $\varphi_\alpha$ è - surgettivo. Se $\varphi_\alpha$ è iniettivo, - si dice che $\alpha$ è \textbf{trascendentale} su $K$ e - $K[x] \cong K[\alpha]$, da cui $[K[\alpha] : K] = + $\varphi_\alpha$ se $K$ è noto, l'omomorfismo completamente + determinato dalla relazione $p \xmapsto{\varphi_\alpha} p(\alpha)$. Si verifica che $\varphi_\alpha$ è + surgettivo. Se $\varphi_\alpha$ è iniettivo, + si dice che $\alpha$ è \textbf{trascendente} su $K$ e + $K[x] \cong K[\alpha]$, da cui $[K[\alpha] : K] = [K[x] : K] = \infty$. Se invece $\varphi_\alpha$ non - è iniettivo, si dice che $\alpha$ è \textbf{algebrico} - su $K$. Si definisce $\mu_\alpha$, detto il \textbf{polinomio + è iniettivo, si dice che $\alpha$ è \textbf{algebrico} + su $K$. Si definisce $\mu_\alpha$, detto il \textbf{polinomio minimo} di $\alpha$ su $K$, il generatore monico - di $\Ker \varphi_\alpha$. IDal momento che $K$ è - in particolare un dominio di integrità, $\mu_\alpha$ è sempre irriducibile. \medskip - - - Si definisce - $\deg_K \alpha := \deg \mu_\alpha$. Se $\alpha$ è - algebrico su $K$, $K[x] / (\mu_\alpha) \cong - K[\alpha]$, e quindi $K[\alpha]$ è un campo. Dacché - $K[\alpha] \subseteq K(\alpha)$, vale allora - $K[\alpha] = K(\alpha)$. Inoltre, poiché $\dim_K K[x] / (\mu_\alpha) = \deg_K \alpha$, vale - anche che $[K(\alpha) : K] = \deg_K \alpha$. - Infine, si verifica che $\alpha$ è algebrico se e solo se - $[K(\alpha) : K]$ è finito. \medskip + di $\Ker \varphi_\alpha$. Dal momento che $K$ è + in particolare un dominio di integrità e che $K[x]$ è un PID, + $\mu_\alpha$ è sempre irriducibile. \smallskip + Si definisce + $\deg_K \alpha := \deg \mu_\alpha$. Se $\alpha$ è + algebrico su $K$, $K[x] / (\mu_\alpha) \cong + K[\alpha]$, e quindi $K[\alpha]$ è un campo. Dacché + $K[\alpha] \subseteq K(\alpha)$, vale allora + $K[\alpha] = K(\alpha)$. Inoltre, poiché $\dim_K K[x] / (\mu_\alpha) = \deg_K \alpha$, vale + anche che $[K(\alpha) : K] = \deg_K \alpha$. + Infine, si verifica che $\alpha$ è algebrico se e solo se + $[K(\alpha) : K]$ è finito. \smallskip - \subsection{Estensioni semplici, algebriche} + \subsection{Estensioni semplici, algebriche} - - Si dice che $L$ è un'\textbf{estensione semplice} di - $K$ se $\exists \alpha \in L$ tale per cui $L = K(\alpha)$. - In tal caso si dice che $\alpha$ è un \textbf{elemento primitivo} di $K$. Si dice che $L$ è un'\textbf{estensione + Si dice che $L$ è un'\textbf{estensione semplice} di + $K$ se $\exists \alpha \in L$ tale per cui $L = K(\alpha)$. + In tal caso si dice che $\alpha$ è un \textbf{elemento primitivo} di $K$. Si dice che $L$ è un'\textbf{estensione algebrica} di $K$ se ogni suo elemento è algebrico su $K$. - Ogni estensione finita è algebrica. Non tutte le - estensioni algebriche sono finite (e.g.~$\overline{\QQ}$ su $\QQ$). \medskip - - - L'insieme degli elementi algebrici di un'estensione - di $K$ su $K$ è un estensione algebrica di $K$. - Pertanto se $\alpha$ e $\beta$ sono algebrici, - $\alpha \pm \beta$, $\alpha \beta$, $\alpha \beta\inv$ - e $\alpha\inv \beta$ (a patto che o $\alpha \neq 0$ o - $\beta \neq 0$) sono algebrici. - - - \subsection{Campi perfetti, estensioni separabili e coniugati} - - - Si dice che un'estensione algebrica $L$ è un'\textbf{estensione separabile} di - $K$ 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, - 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 - di $\mu_\alpha$. Se $K(\alpha)$ è separabile su $K$, - $\alpha$ ha esattamente $\deg_K \alpha$ coniugati, - altrimenti esistono al più $\deg_K \alpha$ coniugati. \medskip - - - Un campo è perfetto se e solo se ha caratteristica - $0$ o altrimenti se l'endomorfismo di - Frobenius è un automorfismo. Equivalentemente, - un campo è perfetto se le derivate dei polinomi - irriducibili sono sempre non nulle. Esempi di - campi perfetti sono allora tutti i campi di - caratteristica $0$ e tutti i campi finiti. - - - \subsection{Campi algebricamente chiusi e chiusura algebrica di $K$} - - Un campo $K$ si dice \textbf{algebricamente chiuso} se - ogni $p \in K[x]$ ammette una radice in $K$. Equivalentemente $K$ è algebricamente chiuso se - ogni $p \in K[x]$ ammette tutte le sue radici in $K$. - Si dice \textbf{chiusura algebrica} di $K$ - una sua estensione algebrica e algebricamente - chiusa. Le chiusure algebriche di $K$ sono - $K$-isomorfe tra loro, e quindi si identifica - con $\overline{K}$ la struttura algebrica della - chiusura algebrica di $K$. \medskip - - - Se $L$ è una sottoestensione di $K$ algebricamente - chiuso, allora $\overline{L}$ è il campo degli - elementi algebrici di $K$ su $L$. Infatti se - $p \in L[x]$, $p$ ammette una radice $\alpha$ in $K$, essendo - algebricamente chiuso. Allora $\alpha$ è un elemento - di $K$ algebrico su $L$, e quindi $\alpha \in \overline{L}$. Per il Teorema fondamentale dell'algebra, - $\overline{\RR} = \CC$. - - - \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 - $K$-immersioni da $K(\alpha)$ in $\overline{K}$ - sono esattamente tante quanti sono i coniugati di - $\alpha$ e sono tali da mappare $\alpha$ ad un suo coniugato. Se $K$ è perfetto, esistono esattamente - $\deg_K \alpha$ $K$-immersioni da $K(\alpha)$ - in $\overline{K}$. \medskip - - - 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 - loro coniugati. \medskip - - - Se $L$ è un'estensione separabile finita, allora per - ogni $\varphi : K \to \overline{K}$ esistono - esattamente $[L : K]$ estensioni $\varphi_i : L \to \overline{K}$ di $\varphi$, ossia omomorfismi - 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 - un'estensione è normale se è il campo di spezzamento - di una famiglia di polinomi (in particolare è il campo - di spezzamento di tutti i polinomi irriducibili che - hanno una radice in $L$). Ancora, un'estensione $L$ - è normale se e solo se per ogni $\alpha \in L$, - i coniugati di $L$ appartengono ancora ad $L$. - Per un'estensione normale, per ogni $K$-immersione - $\varphi : L \to \overline{K}$ si può restringere - il codominio ad un campo isomorfo a $L \subseteq \overline{K}$, e quindi considerare $\varphi$ come - 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 - 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: - - \begin{equation*} + Ogni estensione finita è algebrica. Non tutte le + estensioni algebriche sono finite (e.g.~$\overline{\QQ}$ su $\QQ$). \smallskip + + Gli elementi algebrici di $K$ formano un'estensione algebrica di $K$. + Pertanto se $\alpha$ e $\beta$ sono algebrici, + $\alpha \pm \beta$, $\alpha \beta$, $\alpha \beta\inv$ + e $\alpha\inv \beta$ (a patto che opportunamente $\alpha \neq 0$ o + $\beta \neq 0$) sono algebrici. \smallskip + + Dati $a$, $b \in \QQ$, le estensioni $\QQ(\sqrt{a})$ e + $\QQ(\sqrt{b})$ sono uguali se e solo se $ab$ (o equivalentemente + $a/b$) è un quadrato in $\QQ$. + + \subsection{Campi perfetti, estensioni separabili e coniugati} + + Si dice che un'estensione algebrica $L$ è un'\textbf{estensione separabile} di + $K$ 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, + ossia se ogni polinomio irriducibile è separabile. + In un campo perfetto, ogni estensione algebrica + è separabile. Si definiscono i \textbf{coniugati} di + $\alpha$ algebrico su $K$ come le radici + di $\mu_\alpha$. Se $K(\alpha)$ è separabile su $K$, + $\alpha$ ha esattamente $\deg_K \alpha$ coniugati, + altrimenti esistono al più $\deg_K \alpha$ coniugati. \smallskip + + Un campo è perfetto se e solo se ha caratteristica + $0$ o altrimenti se l'endomorfismo di + Frobenius è un automorfismo. Equivalentemente, + un campo è perfetto se le derivate dei polinomi + irriducibili sono sempre non nulle. Esempi di + campi perfetti sono allora tutti i campi di + caratteristica $0$ e tutti i campi finiti. + + \subsection{Campi algebricamente chiusi e chiusura algebrica di \texorpdfstring{$K$}{K}} + + Un campo $K$ si dice \textbf{algebricamente chiuso} se + ogni $p \in K[x]$ ammette una radice in $K$. Equivalentemente $K$ è algebricamente chiuso se + ogni $p \in K[x]$ ammette tutte le sue radici in $K$. + Si dice \textbf{chiusura algebrica} di $K$ + una sua estensione algebrica e algebricamente + chiusa. Le chiusure algebriche di $K$ sono + $K$-isomorfe tra loro, e quindi si identifica + con $\overline{K}$ la struttura algebrica della + chiusura algebrica di $K$. \smallskip + + Se $L$ è una sottoestensione di $K$ algebricamente + chiuso, allora $\overline{L}$ è il campo degli + elementi algebrici di $K$ su $L$. Infatti se + $p \in L[x]$, $p$ ammette una radice $\alpha$ in $K$, essendo + algebricamente chiuso. Allora $\alpha$ è un elemento + di $K$ algebrico su $L$, e quindi $\alpha \in \overline{L}$. Per il Teorema fondamentale dell'algebra, + $\overline{\RR} = \CC$. + + \subsection{Estensioni normali e di Galois, \texorpdfstring{$K$}{K}-immersioni di un'estensione finita di \texorpdfstring{$K$}{K}} + + Sia $\alpha$ un elemento algebrico su $K$. Allora + $[K(\alpha) : K] = \deg_K \alpha$. Le + $K$-immersioni da $K(\alpha)$ in $\overline{K}$ + sono esattamente tante quanti sono i coniugati di + $\alpha$ e sono tali da mappare $\alpha$ ad un suo coniugato. Infatti, + ogni $K$-immersione si ottiene come passaggio al quoziente + di una mappa $K[x] \to K$ che fissa $K$ il cui nucleo contiene + $\mu_\alpha$. Se $K$ è perfetto, esistono esattamente + $\deg_K \alpha$ $K$-immersioni da $K(\alpha)$ + in $\overline{K}$. \smallskip + + Generalizzando, se $L$ è un'estensione separabile finita, allora per + ogni $\varphi : K \to \overline{K}$ esistono + esattamente $[L : K]$ estensioni $\varphi_i : L \to \overline{K}$ di $\varphi$, ossia omomorfismi + tali per cui $\restr{\varphi_i}{K} = \varphi$. \smallskip + + 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 + loro coniugati. \smallskip + + 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 $\alpha$). \smallskip + + 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 + un'estensione è normale se è il campo di spezzamento + di una famiglia di polinomi (in particolare è il campo + di spezzamento di tutti i polinomi irriducibili che + hanno una radice in $L$). Ancora, un'estensione $L$ + è normale se e solo se per ogni $\alpha \in L$, + i coniugati di $L$ appartengono ancora ad $L$. \smallskip + + Per un'estensione normale ad ogni $K$-immersione + $\varphi : L \to \overline{K}$ si può restringere + il codominio ad $L \subseteq \overline{K}$, e quindi considerare $\varphi$ come + un automorfismo di $L$ che fissa $K$. \smallskip + + Se $\Char K \neq 2$, 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 $K$. \smallskip + + 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 che $L$ è + un'\textbf{estensione di Galois}, e si definisce + 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. \smallskip + + \subsection{Azione di \texorpdfstring{$\Gal(L / K)$}{Gal(L/K)} sulle radici di \texorpdfstring{$L$}{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 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 fedelmente e transitivamente + su $\{a_1, \ldots, a_n\}$ in modo naturale: + + \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)]. + \Gal( & L / K) \curvearrowright S(\{a_1, \ldots, a_n\}) \cong S_n, \\ + & \varphi_i \cdot a_j = \varphi_i(a_j). \end{split} - \end{equation*} - - 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 + \end{equation*} + + Poiché l'azione è fedele, vale che + $\Gal(L / K) \mono S_n$, e quindi $\abs{\Gal(L / K)} \mid n!$. + Se $\Gal(L / K)$ è abeliano + (e in tal caso si dice che $L$ è un'\textbf{estensione abeliana}), l'azione + è anche libera, e quindi per il Lemma orbita-stabilizzatore vale + $\abs{\Gal(L / K)} = n$. \smallskip + + Inoltre, poiché $[K(a_1) : K] = n$, vale in particolare: + \[ n \mid \abs{\Gal(L / K)} = [L : K] \mid n!. \] + + \section{Proprietà su torri, traslato, composto e intersezione} + + Sia $\mathcal{P}$ una proprietà. Allora si + studia la proprietà $\mathcal{P}$ secondo + le seguenti quattro 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 $L / K$, allora vale anche per $LM / M$, ossia + vale sul ramo parallelo a quello di $L / K$, + \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. \smallskip + + 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. \smallskip + + La seguente tabella raccoglie le proprietà + delle estensioni: + \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 + $L = K(a + bt)$ (perché ha il numero giusto di coniugati per + generare tutto $L$!). Reiterando questo algoritmo su + tutti i generatori dell'estensione, si ottiene + un elemento primitivo desiderato. + + \section{Corrispondenza di Galois} + + \subsection{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 $L$ composta + dagli elementi di $L$ + fissati da tutti gli elementi di $H$. + In particolare vale che $L^H = K \iff H = \Gal(L / K)$. + Conseguentemente, vale il seguente teorema: + + \begin{theorem}[di corrispondenza di Galois] + 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: + la cui inversa $\beta : \mathcal{G} \to \mathcal{E}$ + è tale per cui: + \[ H \xmapsto{\beta} L^H \subseteq L. \] + Questa corrispondenza manda sottoestensioni di grado $n$ su $K$ + in sottogruppi di indice $n$ e viceversa (infatti $[F : K] = [L : K] / [L : F] = \abs{\Gal(L / K)} / \abs{\Gal(L : F)} = + [\Gal(L / K) : \Gal(L / F)]$). \smallskip + + 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: + \[ \Gal(F / K) \cong \faktor{\Gal(L / K)}{\Gal(L / F)}. \] + Infine valgono le seguente 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 $L^H \subseteq L^Q \iff H \geq Q$ (la corrispondenza inverte le inclusioni), \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] + \end{theorem} + + 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: + \[\begin{tikzcd}[row sep=small] {\{e\}} \\ \\ H \\ @@ -600,76 +565,113 @@ \arrow[no head, from=1-1, to=3-1] \arrow[no head, from=3-1, to=5-1] \end{tikzcd}\] - - \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 $\QQ$}, \\ + + \subsection{Relazioni tra gruppi di Galois in un diagramma} + + Consideriamo il seguente diagramma di campi: + \[\begin{tikzcd} + & LM \\ + L & {} & M \\ + & {L \cap M} \\ + & K + \arrow[no head, from=4-2, to=3-2] + \arrow[no head, from=3-2, to=2-1] + \arrow[no head, from=2-1, to=1-2] + \arrow[no head, from=3-2, to=2-3] + \arrow[no head, from=2-3, to=1-2] + \end{tikzcd}\] + + Allora $\Gal(LM/M)$ è sempre isomorfo a $\Gal(L/(L \cap M))$ tramite: + \[ + \varphi \mapsto \restr{\varphi}{L}. + \] + Per la validità dell'isomorfismo, è sufficiente che $L$ sia di Galois, ma non + è necessario che lo sia anche $M$ (benché debba essere finita). \smallskip + + Pertanto, se $L \cap M = K$, vale $\Gal(LM/M) \cong \Gal(L/K)$, e quindi + in tal caso: + \[ + [LM : K] = [L : K] [M : K]. + \] + + Sussiste sempre un'immersione: + \[ + \Gal(LM/K) \hookrightarrow \Gal(L/K) \times \Gal(M/K), + \] + che manda $\varphi$ in $(\restr{\varphi}{L}, \restr{\varphi}{M})$. Tale + immersione è un isomorfismo se e solo se $L \cap M = K$, per il precedente + risultato. + + \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 $\FF_{p^n}$. \smallskip + + 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 $\alpha$. \smallskip + + Se il campo base non è $\FF_p$, si può utilizzare la corrispondenza + di Galois per determinare comunque che: + \[ + \Gal(\FF_{p^n} / \FF_{p^m}) \cong \faktor{\ZZ / n \ZZ}{\ZZ / m \ZZ}. + \] + + \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: + + \[ \Gal(L / \QQ) \cong \begin{cases} + \ZZmod{4} & \se b \text{ è quadrato in $\QQ$}, \\ \ZZmod{2} \times \ZZmod{2} & \se b \Delta \text{ è quadrato in $\QQ$}, \\ - D_4 & \altrimenti. + D_4 & \altrimenti. \end{cases} \] - - - \subsection{Radici di primi in $\QQ$} - - Siano $p_1$, ..., $p_n$ numeri primi distinti. - Allora - vale che: - \[ \Gal(\QQ(\sqrt{p_1}, \ldots, \sqrt{p_n})/\QQ) \cong (\ZZmod{2})^n. \] - - \subsection{I polinomi ciclotomici $\Phi_n(x)$} - - Sia $\Phi_n(x)$ l'$n$-esimo polinomio ciclotomico, così definito: - \[ \Phi_n(x) = \prod_{\substack{1 \leq d \leq n \\ \MCD(d, n) = 1}} (x - \zeta_n^d), \] - dove $\zeta_n$ è una radice primitiva $n$-esima dell'unità. \medskip - - - Tale polinomio è sempre a coefficienti interi ed è inoltre primitivo - su $\ZZ[x]$. Vale inoltre che: - \[ x^n - 1 = \prod_{m \mid n} \Phi_m(x). \] - Il campo di spezzamento di $\Phi_n(x)$ su $\QQ$ è - $\QQ(\zeta_n)$, che è un'estensione normale, separabile e finita, - e pertanto di Galois. \medskip - - - Inoltre vale che: - \[ \Gal(\QQ(\zeta_n)/\QQ) \cong (\ZZmod{n})^\times, \] - e dunque $\Phi_n(x)$ è sempre irriducibile su $\QQ$. - - \vfill - \hrule - ~\\ - Ad opera di Gabriel Antonio Videtta, \url{https://poisson.phc.dm.unipi.it/~videtta/}. - ~\\Reperibile su - \url{https://notes.hearot.it}, nella sezione \textit{Secondo anno $\to$ Algebra 1 $\to$ 3. Teoria delle estensioni di campo e di Galois $\to$ Scheda riassuntiva di Teoria dei campi e di Galois}. - \end{multicols} - + + \subsection{Radici di primi in \texorpdfstring{$\QQ$}{ℚ}} + + Siano $p_1$, ..., $p_n$ numeri primi distinti. + Allora vale che: + \[ \Gal(\QQ(\sqrt{p_1}, \ldots, \sqrt{p_n})/\QQ) \cong (\ZZmod{2})^n. \] + + \subsection{I polinomi ciclotomici \texorpdfstring{$\Phi_n(x)$}{Φₙ(x)}} + + Sia $\Phi_n(x)$ l'$n$-esimo polinomio ciclotomico, così definito: + \[ \Phi_n(x) = \prod_{\substack{1 \leq d \leq n \\ \MCD(d, n) = 1}} (x - \zeta_n^d), \] + dove $\zeta_n$ è una radice primitiva $n$-esima dell'unità. \smallskip + + Tale polinomio è sempre a coefficienti interi ed è inoltre primitivo + su $\ZZ[x]$. Vale inoltre che: + \[ x^n - 1 = \prod_{m \mid n} \Phi_m(x). \] + Il campo di spezzamento di $\Phi_n(x)$ su $\QQ$ è + $\QQ(\zeta_n)$, che è un'estensione normale, separabile e finita, + e pertanto di Galois. \smallskip + + Inoltre vale che: + \[ \Gal(\QQ(\zeta_n)/\QQ) \cong (\ZZmod{n})^\times, \] + e dunque $\Phi_n(x)$ è sempre irriducibile su $\QQ$. \smallskip + + \vfill + \hrule + ~\\ + Ad opera di Gabriel Antonio Videtta, \url{https://poisson.phc.dm.unipi.it/~videtta/}. + ~\\Reperibile su + \url{https://github.com/hearot/notes}, nella sezione \textit{Corsi $\to$ Algebra 1 $\to$ 3. Teoria delle estensioni di campo e di Galois $\to$ Scheda riassuntiva di Teoria dei campi e di Galois}. +\end{multicols} + \end{document} \ No newline at end of file diff --git a/Corsi/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/notes_2023.sty b/Corsi/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/notes_2023.sty new file mode 100644 index 0000000..f31fc38 --- /dev/null +++ b/Corsi/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/notes_2023.sty @@ -0,0 +1,451 @@ +\ProvidesPackage{notes_2023} + +\usepackage{amsmath,amssymb} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{amssymb} +\usepackage{amsopn} +\usepackage{bookmark} +\usepackage{faktor} +\usepackage[utf8]{inputenc} +\usepackage{mathtools} +\usepackage{nicefrac} +\usepackage{stmaryrd} +\usepackage{marvosym} +\usepackage{float} +\usepackage{enumerate} +\usepackage{scalerel} +\usepackage{stackengine} +\usepackage{wasysym} + +\usepackage{tikz-cd} + +\usepackage{quiver} + +\usepackage[italian]{babel} + +\usepackage{tabularx} + +% Setup preliminari +\setlength{\extrarowheight}{4pt} + +\newcommand{\system}[1]{\begin{cases} #1 \end{cases}} + +\newcommand{\wip}{\begin{center}\textit{Questo avviso sta ad indicare che questo documento è ancora una bozza e non è + da intendersi né completo, né revisionato.}\end{center}} + +\newcommand\hr{\vskip 0.05in \par\vspace{-.5\ht\strutbox}\noindent\hrulefill\par} + +% Modalità matematica/fisica +\newcommand{\SMatrix}[1]{\begin{psmallmatrix}#1\end{psmallmatrix}} + +\let\oldvec\vec +\renewcommand{\vec}[1]{\underline{#1}} + +\newcommand{\con}{\text{con }} +\newcommand{\dove}{\text{dove }} +\newcommand{\E}{\text{ e }} +\newcommand{\altrimenti}{\text{altrimenti}} +\newcommand{\se}{\text{se }} +\newcommand{\tc}{\text{ t.c. }\!} +\newcommand{\epari}{\text{ è pari}} +\newcommand{\edispari}{\text{ è dispari}} + +\newcommand{\nl}{\ \\} + +\newcommand{\bigmid}{\;\middle\vert\;} + +\newcommand{\lemmaref}[1]{\textit{Lemma \ref{#1}}} +\newcommand{\thref}[1]{\textit{Teorema \ref{#1}}} + +\newcommand{\li}[0]{$\blacktriangleright\;\;$} + +\newcommand{\tends}[1]{\xrightarrow[\text{$#1$}]{}} +\newcommand{\tendsto}[1]{\xrightarrow[\text{$x \to #1$}]{}} +\newcommand{\tendstoy}[1]{\xrightarrow[\text{$y \to #1$}]{}} +\newcommand{\tendston}[0]{\xrightarrow[\text{$n \to \infty$}]{}} + +\setlength\parindent{0pt} + +% Principio di induzione e setup dimostrativi. +\newcommand{\basestep}{\mbox{(\textit{passo base})}\;} +\newcommand{\inductivestep}{\mbox{(\textit{passo induttivo})}\;} + +\newcommand{\rightproof}{\mbox{($\implies$)}\;} +\newcommand{\leftproof}{\mbox{($\impliedby$)}\;} + +% Spesso utilizzati al corso di Fisica 1. +\newcommand{\dx}{\dot{x}} +\newcommand{\ddx}{\ddot{x}} +\newcommand{\dv}{\dot{v}} + +\newcommand{\del}{\partial} +\newcommand{\tendstot}[0]{\xrightarrow[\text{$t \to \infty$}]{}} + +\newcommand{\grad}{\vec{\nabla}} +\DeclareMathOperator{\rot}{rot} + +\newcommand{\ihat}{\hat{i}} +\newcommand{\jhat}{\hat{j}} +\newcommand{\khat}{\hat{k}} + +\newcommand{\der}[1]{\frac{d#1}{dx}} +\newcommand{\parx}{\frac{\del}{\del x}} +\newcommand{\pary}{\frac{\del}{\del y}} +\newcommand{\parz}{\frac{\del}{\del z}} + +% Spesso utilizzati al corso di Analisi 1. +%\newcommand{\liminf}{\lim_{x \to \infty}} +\newcommand{\liminfty}{\lim_{x \to \infty}} +\newcommand{\liminftym}{\lim_{x \to -\infty}} +\newcommand{\liminftyn}{\lim_{n \to \infty}} +\newcommand{\limzero}{\lim_{x \to 0}} +\newcommand{\limzerop}{\lim_{x \to 0^+}} +\newcommand{\limzerom}{\lim_{x \to 0^-}} + +\newcommand{\xbar}{\overline{x}} +\newcommand{\ybar}{\overline{y}} +\newcommand{\tbar}{\overline{t}} +\newcommand{\zbar}{\overline{z}} +\newcommand{\RRbar}{\overline{\RR}} + +% Spesso utilizzati al corso di Geometria 2 + +\newcommand{\PP}{\mathbb{P}} +\newcommand{\PPGL}{\mathbb{P}\!\GL} + +% Spesso utilizzati al corso di Geometria 1. + +\DeclareMathOperator{\OO}{O} % gruppo ortogonale +\DeclareMathOperator{\SOO}{SO} % gruppo ortogonale speciale + +\newcommand{\proj}[1]{\Matrix{#1 \\[0.03in] \hline 1}} +\newcommand{\projT}[1]{\Matrix{#1 & \rvline & 1}^\top} + +\newcommand{\cc}{\mathcal{C}} + +\let\AA\undefined +\DeclareMathOperator{\Iso}{Iso} +\newcommand{\AA}{\mathcal{A}} +\newcommand{\MM}{\mathcal{M}} +\newcommand{\KKxn}{\KK[x_1, \ldots, x_n]} + +\let\ext\faktor +\newcommand{\quot}[1]{/{#1}} + +\newcommand{\Aa}{\mathcal{A}} +\newcommand{\Ad}[1]{\mathcal{A}_{#1}} +\DeclareMathOperator{\An}{\mathcal{A}_n} +\DeclareMathOperator{\AnK}{\mathcal{A}_n(\KK)} +\DeclareMathOperator{\Giac}{Giac} + +\DeclareMathOperator{\IC}{IC} +\DeclareMathOperator{\Aff}{Aff} +\DeclareMathOperator{\Orb}{Orb} +\DeclareMathOperator{\Stab}{Stab} +\DeclareMathOperator{\Gr}{Gr} +\newcommand{\vvec}[1]{\overrightarrow{#1}} + +\newcommand{\conj}[1]{\overline{#1}} + +\DeclareMathOperator{\PH}{PH} +\DeclareMathOperator{\PS}{PS} + +\let\imm\Im +\let\Im\undefined +\DeclareMathOperator{\Im}{Im} +\DeclareMathOperator{\Rad}{Rad} +\newcommand{\restr}[2]{ + #1\arrowvert_{#2} +} + +\newcommand{\innprod}[1]{\langle #1 \rangle} + +\newcommand{\zerovecset}{\{\vec 0\}} +\newcommand{\bigzero}{\mbox{0}} +\newcommand{\rvline}{\hspace*{-\arraycolsep}\vline\hspace*{-\arraycolsep}} + +\newcommand{\Idv}{\operatorname{Id}_V} +\newcommand{\Idw}{\operatorname{Id}_W} +\newcommand{\IdV}[1]{\operatorname{Id}_{#1}} +\DeclareMathOperator{\CI}{CI} +\DeclareMathOperator{\Bil}{Bil} +\DeclareMathOperator{\Mult}{Mult} +\DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\Ann}{Ann} +\DeclareMathOperator{\adj}{adj} +\DeclareMathOperator{\Cof}{Cof} +\DeclareMathOperator{\pr}{pr} +\DeclareMathOperator{\Sp}{sp} + +\newcommand{\dperp}{{\perp\perp}} + +\newcommand{\Eigsp}[0]{V_{\lambda}} +\newcommand{\Gensp}[0]{\widetilde{V_{\lambda}}} +\newcommand{\eigsp}[1]{V_{\lambda_{#1}}} +\newcommand{\gensp}[1]{\widetilde{V_{\lambda_{#1}}}} +\newcommand{\genspC}[1]{\widetilde{V_{#1}}} + +\DeclareMathOperator{\val}{val} +\DeclareMathOperator{\Span}{Span} +\newcommand{\charpoly}[1]{p_{#1}} +\newcommand{\charpolyrestr}[2]{p_{#1\arrowvert_#2}\hspace{-1pt}(\lambda)} +\newcommand{\minpoly}[1]{\varphi_{#1}} +\newcommand{\valf}{\val_f} +\newcommand{\valfv}{\val_{f,\V}} +\newcommand{\e}[1]{\vec{e_{#1}}} +\newcommand{\V}{\vec{v}} +\newcommand{\VV}[1]{\vec{v_{#1}}} +\newcommand{\basisdual}{\dual{\basis}} 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+\newcommand{\mono}{\hookrightarrow} + +\newcommand{\pev}{\nu_p} +\newcommand{\exactdiv}{\mathrel\Vert} +\newcommand{\pset}{\mathcal{P}} + +\newcommand{\Dn}{D_n} +\newcommand{\Sn}{S_n} + +\newcommand{\mulgrp}[1]{\left(#1\right)^*} + +\newcommand{\ZZmulmod}[1]{\mulgrp{\ZZmod{#1}}} + +\newcommand{\bij}{\leftrightarrow} +\newcommand{\ZZpmod}[1]{\ZZ \quot {\left(#1\right)} \ZZ} +\newcommand{\ZZmod}[1]{\ZZ \quot #1 \ZZ} +\newcommand{\cleq}[1]{\overline{#1}} + +\newcommand{\rotations}{\mathcal{R}} +\newcommand{\gen}[1]{\langle #1 \rangle} +\DeclareMathOperator{\Cl}{Cl} + +\newcommand{\actson}{\circlearrowleft} +\newcommand{\Cyc}[1]{\left<#1\right>} + +% Comandi personali. + +\newcommand{\card}[1]{\left|#1\right|} +\newcommand{\nsqrt}[2]{\!\sqrt[#1]{#2}\,} +\newcommand{\zeroset}{\{0\}} +\newcommand{\setminuszero}{\setminus \{0\}} + +\newenvironment{solution} +{\textit{Soluzione.}\,} + +\theoremstyle{definition} + +\let\abstract\undefined +\let\endabstract\undefined + +\newtheorem*{abstract}{Abstract} +\newtheorem*{algorithm}{Algoritmo} +\newtheorem*{corollary}{Corollario} +\newtheorem*{definition}{Definizione} +\newtheorem*{example}{Esempio} +\newtheorem{exercise}{Esercizio} +\newtheorem{lemma}{Lemma} +\newtheorem*{nlemma}{Lemma} +\newtheorem*{note}{Nota} +\newtheorem*{remark}{Osservazione} +\newtheorem*{proposition}{Proposizione} +\newtheorem*{summary}{Sommario} +\newtheorem*{theorem}{Teorema} +\newtheorem*{scheme}{Schema della dimostrazione} + +\newcommand{\basis}{\mathcal{B}} +\newcommand{\BB}{\mathcal{B}} + +\newcommand{\basisC}{\mathcal{B}} + +\newcommand{\HH}{\mathbb{H}} + +\newcommand{\FFp}[1]{\mathbb{F}_p} +\newcommand{\FFpx}[1]{\mathbb{F}_p[x]} + +\newcommand{\CCx}{\mathbb{C}[x]} + +\newcommand{\KK}{\mathbb{K}} +\newcommand{\KKx}{\mathbb{K}[x]} + +\newcommand{\QQx}{\mathbb{Q}[x]} +\newcommand{\RRx}{\mathbb{R}[x]} + +\newcommand{\ZZi}{\mathbb{Z}[i]} +\newcommand{\ZZp}{\mathbb{Z}_p} +\newcommand{\ZZpx}{\mathbb{Z}_p[x]} +\newcommand{\ZZx}{\mathbb{Z}[x]} + +\newcommand{\ii}{\mathbf{i}} +\newcommand{\jj}{\mathbf{j}} +\newcommand{\kk}{\mathbf{k}} + +\newcommand{\bidual}[1]{#1^{**}} +\newcommand{\dual}[1]{#1^{*}} +\newcommand{\LL}[2]{\mathcal{L} \left(#1, \, #2\right)} % L(V, W) +\newcommand{\Ll}{\mathcal{L}} + +\newcommand{\nsg}{\triangleleft} % sottogruppo normale proprio +\newcommand{\nsgeq}{\trianglelefteqslant} % sottogruppo normale + +% evan.sty original commands +\newcommand{\cbrt}[1]{\sqrt[3]{#1}} +\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} +\newcommand{\ceiling}[1]{\left\lceil #1 \right\rceil} +\newcommand{\mailto}[1]{\href{mailto:#1}{\texttt{#1}}} +\newcommand{\eps}{\varepsilon} +\newcommand{\vocab}[1]{\textbf{\color{blue}\sffamily #1}} +\providecommand{\alert}{\vocab} +\newcommand{\catname}{\mathsf} +\providecommand{\arc}[1]{\wideparen{#1}} + +% From H113 "Introduction to Abstract Algebra" at UC Berkeley +\newcommand{\CC}{\mathbb C} +\newcommand{\FF}{\mathbb F} +\newcommand{\NN}{\mathbb N} +\newcommand{\QQ}{\mathbb Q} +\newcommand{\RR}{\mathbb R} +\newcommand{\ZZ}{\mathbb Z} +\DeclareMathOperator{\Aut}{Aut} +\DeclareMathOperator{\Inn}{Inn} +\DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Gal}{Gal} +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\SL}{SL} + +% From Kiran Kedlaya's "Geometry Unbound" +\newcommand{\abs}[1]{\left\lvert #1 \right\rvert} +\newcommand{\norm}[1]{\left\lVert #1 \right\rVert} +\newcommand{\dang}{\measuredangle} %% Directed angle +\newcommand{\ray}[1]{\overrightarrow{#1}} +\newcommand{\seg}[1]{\overline{#1}} + +% From M275 "Topology" at SJSU +\newcommand{\Id}{\mathrm{Id}} +\newcommand{\id}{\mathrm{id}} +\newcommand{\taking}[1]{\xrightarrow{#1}} +\newcommand{\inv}{^{-1}} + +\DeclareMathOperator{\ord}{ord} +\newcommand{\defeq}{\overset{\mathrm{def}}{=}} +\newcommand{\defiff}{\overset{\mathrm{def}}{\iff}} + +% From the USAMO .tex files +\newcommand{\dg}{^\circ} + +\newcommand{\liff}{\leftrightarrow} +\newcommand{\lthen}{\rightarrow} +\newcommand{\opname}{\operatorname} +\newcommand{\surjto}{\twoheadrightarrow} +\newcommand{\injto}{\hookrightarrow} + +% Alcuni degli operatori più comunemente utilizzati. + +\DeclareMathOperator{\Char}{char} +\DeclareMathOperator{\Dom}{Dom} +\DeclareMathOperator{\Fix}{Fix} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\existsone}{\exists !} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Imm}{Imm} +\DeclareMathOperator{\Ker}{ker} +\DeclareMathOperator{\rank}{rank} +\DeclareMathOperator{\MCD}{MCD} +\DeclareMathOperator{\Mor}{Mor} +\DeclareMathOperator{\mcm}{mcm} +\DeclareMathOperator{\Sym}{Sym} +\DeclareMathOperator{\tr}{tr} + +% Reimposta alcuni simboli presenti di default in LaTeX con degli analoghi +% più comuni. + +\let\oldemptyset\emptyset +\let\emptyset\varnothing + +% Trasforma alcuni simboli in operatori matematici. + +\let\oldcirc\circ +\let\circ\undefined +\DeclareMathOperator{\circ}{\oldcirc} + +\let\oldexists\exists +\let\exists\undefined +\DeclareMathOperator{\exists}{\oldexists} + +\let\oldforall\forall +\let\forall\undefined +\DeclareMathOperator{\forall}{\oldforall} + +\let\oldnexists\nexists +\let\nexists\undefined +\DeclareMathOperator{\nexists}{\oldnexists} + +\let\oldland\land +\let\land\undefined +\DeclareMathOperator{\land}{\oldland} + +\let\oldlnot\lnot +\let\lnot\undefined +\DeclareMathOperator{\lnot}{\oldlnot} + +\let\oldlor\lor +\let\lor\undefined +\DeclareMathOperator{\lor}{\oldlor} + +\DeclareOption{counter}{ + \let\algorithm\@undefined + \let\endalgorithm\@undefined + \let\corollary\@undefined + \let\endcorollary\@undefined + \let\c@lemma\@undefined + \let\lemma\@undefined + \let\endlemma\@undefined + \let\proposition\@undefined + \let\endproposition\@undefined + \let\theorem\@undefined + \let\endtheorem\@undefined + + \newtheorem{algorithm}{Algoritmo}[chapter] + \newtheorem{corollary}{Corollario}[chapter] + \newtheorem{lemma}{Lemma}[chapter] + \newtheorem{proposition}{Proposizione}[chapter] + \newtheorem{theorem}{Teorema}[chapter] +} + +\ProcessOptions\relax + +\author{di Gabriel Antonio Videtta} +\date{\vspace{-0.5cm}}