From b678f72a50fde8e93ff74f1beab067d793e84eec Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 12 Jan 2023 15:58:45 +0100 Subject: [PATCH] feat(algebrario): aggiunge nuovo capitolo e aggiorna evan.sty --- ... Introduzione alla teoria degli anelli.tex | 1 + .../10. Introduzione a teoria dei campi.tex | 285 +++++ ...1. Teoremi rilevanti sui campi finiti.tex} | 4 +- .../3. Esempi notevoli di anelli euclidei.tex | 12 +- ...ondamentali di Z[i], Zp[x], Z[x], Q[x].tex | 4 +- .../5. Irriducibilità in Z[x] e Q[x].tex | 6 +- ... di K[x] e delle estensioni algebriche.tex | 6 +- Aritmetica/7. Estensioni algebriche di K.tex | 2 +- Aritmetica/8. Campi di spezzamento.tex | 24 +- ...entale dell'algebra ed estensioni di Q.tex | 2 +- Aritmetica/aritmetica.pdf | Bin 807444 -> 846031 bytes Aritmetica/aritmetica.tex | 100 +- Aritmetica/bibliography.bib | 46 +- Aritmetica/evan.sty | 1089 ++++++----------- 14 files changed, 724 insertions(+), 857 deletions(-) create mode 100644 Aritmetica/10. Introduzione a teoria dei campi.tex rename Aritmetica/{10. Teoremi rilevanti sui campi finiti.tex => 11. Teoremi rilevanti sui campi finiti.tex} (96%) diff --git a/Aritmetica/1. Introduzione alla teoria degli anelli.tex b/Aritmetica/1. Introduzione alla teoria degli anelli.tex index 95e46ac..679ca27 100644 --- a/Aritmetica/1. Introduzione alla teoria degli anelli.tex +++ b/Aritmetica/1. Introduzione alla teoria degli anelli.tex @@ -309,6 +309,7 @@ Adesso è possibile enunciare il seguente fondamentale teorema: \end{proof} \begin{corollary} + \label{cor:primo_isomorfismo_iniettivo} Sia $\phi : A \to B$ un monomorfismo. $A \cong \Imm \phi$. \end{corollary} diff --git a/Aritmetica/10. Introduzione a teoria dei campi.tex b/Aritmetica/10. Introduzione a teoria dei campi.tex new file mode 100644 index 0000000..512342b --- /dev/null +++ b/Aritmetica/10. Introduzione a teoria dei campi.tex @@ -0,0 +1,285 @@ +\section{Introduzione alla teoria dei campi} + +\subsection{La caratteristica di un campo} + +Si consideri il seguente omomorfismo: + +\[ \psi : \ZZ \to \KK, \] + +\vskip 0.1in + +completamente determinato dalla condizione $\psi(1) = 1$, dacché +$\ZZ$ è generato da $1$. Si studia innanzitutto il caso in cui +$\Ker \psi = (0)$. In questo caso, $\psi$ è un monomorfismo, e per +il \corref{cor:primo_isomorfismo_iniettivo}, $\ZZ \cong \Imm \psi$. \\ + +Pertanto, $\KK$ ammetterebbe come sottoanello una copia isomorfa di $\ZZ$. +Inoltre, poiché $\KK$ è un campo, deve anche ammetterne gli inversi, e quindi +ammetterebbe come sottocampo una copia isomorfa di $\QQ$. La seguente +definizione classificherà questi tipi di campo. \\ + +\begin{definition} + Si dice che un campo $\KK$ è di \textbf{caratteristica zero} ($\Char \KK = 0$), + quando $\Ker \psi = (0)$. +\end{definition} + +Altrimenti, se $\Ker \psi \neq (0)$, dacché $\ZZ$ è un anello euclideo, +$\Ker \psi$ deve essere monogenerato da un intero $n$, ossia $\Ker \psi = (n)$. \\ + +Tuttavia non tutti gli interi sono ammissibili. Sia infatti $n$ non primo, allora +$n = ab$ con $a$, $b \neq \pm 1$. Si nota innanzitutto che $\psi(a) \neq 0$, +se infatti fosse nullo, $n$ dovrebbe dividere $a$, impossibile dal momento +che $\card{a} < \card{n}$, \Lightning{}. Analogamente anche $\psi(b) \neq 0$. \\ + +Se $n$ fosse generatore di $\Ker \psi$ si ricaverebbe allora che: + +\[ \underbrace{\psi(a)}_{\neq\,0} \underbrace{\psi(b)}_{\neq\,0} = \psi(n) = 0, \] + +\vskip 0.1in + +che è assurdo, dal momento che $\KK$, in quanto campo, è anche un dominio. +Quindi $n$ deve essere un numero primo. In particolare, allora, per +il \nameref{th:primo_isomorfismo}, $\ZZp = \ZZ/(p) \cong \Imm \psi$, +ossia $\KK$ contiene una copia isomorfa di $\ZZp$, a cui ci riferiremo +semplicemente con $\FFpp$. \\ + +Allora, poiché sia $\KK$ che $\FFpp$ sono campi, $\KK$ è uno spazio +vettoriale su $\FFpp$. Si può dunque classificare quest'ultimo tipo di +campi con la seguente definizione: + +\begin{definition} + Si dice che un campo $\KK$ è di \textbf{caratteristica $p$} ($\Char \KK = p$) + quando $\Ker \psi = (p)$, con $p$ primo. +\end{definition} + +\begin{remark*} + La caratteristica di un campo \textbf{non} distingue i campi finiti + dai campi infiniti. Esistono infatti campi infiniti di caratteristica + $p$, come il campo delle funzioni razionali su $\ZZp$: + + \[ \ZZ_p(x) = \left\{ \frac{f(x)}{g(x)} \mid f(x),\, g(x) \in \ZZpx,\, g(x) \neq 0 \right\}. \] + + \vskip 0.1in + + Infatti $\psi(p) = p \, \psi(1) = 0$. +\end{remark*} + +\subsection{Prime proprietà dei campi di caratteristica \texorpdfstring{$p$}{p}} + +Come si è appena visto, un campo $\KK$ di caratteristica $p$ contiene +al suo interno un sottocampo $\FFpp$ isomorfo a $\ZZp$, ed è per questo +uno spazio vettoriale su di esso. A partire da questa informazione si +può dimostrare la seguente proposizione. + +\begin{proposition} + \label{prop:campo_char_p_prodotto_per_p} + Sia $\KK$ un campo di caratteristica $p$. Allora, per ogni + elemento $v$ di $\KK$, $pv=0$. +\end{proposition} + +\begin{proof} + Considerando ogni elemento di $\KK$ come vettore e $p$ come + scalare, si ricava che: + + \[ pv=(\underbrace{1+\ldots+1}_{p\text{ volte}})v= + (\underbrace{\psi(1)+\ldots+\psi(1)}_{p\text{ volte}})v= + \psi(p)v=0v=0. \] + +\end{proof} + +Mentre, partendo da questa proposizione, si può dimostrare il +seguente teorema. + +\begin{theorem}[\textit{Teorema del binomio ingenuo}] + \label{th:binomio_ingenuo} + Siano $a$ e $b$ elementi di un campo di caratteristica $p$. Allora + $(a+b)^p = a^p + b^p$. +\end{theorem} + +\begin{proof} + Per dimostrare la tesi si applica la formula del binomio di Newton + nel seguente modo: + + \[ (a+b)^p = \sum_{i=0}^p \binom{p}{i} a^{p-i}b^p. \] + + \vskip 0.1in + + Tuttavia, dal momento che $p$ è un fattore di tutti i binomiali per + $1 \leq i \leq p-1$, tutti i termini computati con queste $i$ + sono nulli per la \propref{prop:campo_char_p_prodotto_per_p}. + Si desume così l'identità della tesi. +\end{proof} + +\subsection{L'omomorfismo di Frobenius} + +\begin{definition} + Dato un campo $\KK$ di caratteristica $p$, si definisce + \textbf{omomorfismo di Frobenius} per il campo $\KK$ + la funzione: + + \[ \Frob : \KK \to \KK,\, a \mapsto a^p. \] +\end{definition} + +\begin{remark*} + In effetti, l'omomorfismo di Frobenius è un omomorfismo. \\ + + Infatti, $\Frob(1) = 1^p = 1$. Inoltre tale funzione + rispetta la linearità per il \nameref{th:binomio_ingenuo}: + + \[ \Frob(a + b) = (a+b)^p = a^p + b^p = \Frob(a) + \Frob(b), \] + + \vskip 0.1in + + e chiaramente anche la moltiplicatività: + + \[ \Frob(ab) = (ab)^p = a^p b^p = \Frob(a) \Frob(b). \] +\end{remark*} + +\begin{proposition} + \label{prop:frobenius_monomorfismo} + L'omomorfismo di Frobenius di un campo $\KK$ di caratteristica + $p$ è un monomorfismo. +\end{proposition} + +\begin{proof} + Si prenda in considerazione $\Ker \Frob$. Esso è sicuramente + un ideale diverso da $\KK$, dacché $1 \notin \Ker \Frob$. + Tuttavia, se $\Ker \Frob \neq (0)$, $\Ker \Frob$, dal + momento che $\KK$, in quanto campo, è un anello euclideo, + e quindi un PID, è monogenerato da un invertibile. \\ + + Se però così fosse, $\Ker \Frob$ coinciderebbe con il + campo $\KK$ stesso, \Lightning{}. Quindi $\Ker \Frob = (0)$, + da cui la tesi. +\end{proof} + +\begin{proposition} + Sia $\KK$ un campo finito di caratteristica $p$. Allora + l'omomorfismo di Frobenius è un automorfismo. +\end{proposition} + +\begin{proof} + Dalla \propref{prop:frobenius_monomorfismo} è noto che + $\Frob$ sia già un monomorfismo. Dal momento che + il dominio e il codominio sono lo stesso e constano + entrambi dunque di un numero finito di elementi, + se $\Frob$ non fosse surgettivo, vi sarebbe un elemento + di $\KK$ a cui non è associato nessun elemento di $\KK$ + mediante $\Frob$. \\ + + Per il principio dei cassetti, allora, spartendo + $\card{\KK}$ elementi in $\card{\KK}-1$ elementi, + vi sarebbe almeno un elemento dell'immagine a cui + sarebbero associati due elementi del dominio. Tuttavia + questo è assurdo dal momento che $\Frob$ è un + monomorfismo. Quindi $\Frob$ è un epimorfismo. \\ + + Dacché $\Frob$ è contemporaneamente un endomorfismo, + un monomorfismo e un epimorfismo, è allora anche + un automorfismo. +\end{proof} + +\begin{proposition} + \label{prop:punti_fissi_frobenius_campo} + Sia $\KK$ un campo di caratteristica $p$ e si + definisca l'insieme dei punti fissi del suo + omomorfismo di Frobenius: + + \[ \Fix(\Frobexp^n) = \{ a \in \KK \mid \Frobexp^n(a) = a \} .\] + + \vskip 0.1in + + Allora $\Fix(\Frobexp^n)$ è un sottocampo di $\KK$. +\end{proposition} + +\begin{proof} + Affinché $\Fix(\Frobexp^n)$ sia un sottocampo di $\KK$, + la sua somma e la sua moltiplicazione devono essere ben + definite, e ogni suo elemento deve ammettere un inverso + sia additivo che moltiplicativo. \\ + + Siano allora $a$, $b \in \Fix(\Frobexp^n)$. + $\Frobexp^n$ è un omomorfismo, in quanto è composizione + di omomorfismi (in particolare, dello stesso omomorfismo + $\Frobexp$). Sfruttando le proprietà + degli omomorfismi si dimostra dunque + che $a+b \in \Fix(\Frobexp^n)$: + + \[ \Frobexp^n(a+b) = \Frobexp^n(a) + \Frobexp^n(b) = a+b, \] + + \vskip 0.1in + + e che $ab \in \Fix(\Frobexp^n)$: + + \[ \Frobexp^n(ab) = \Frobexp^n(a)\Frobexp^n(b) = ab. \] + + \vskip 0.1in + + Analogamente si dimostra che $-a \in \Fix(\Frobexp^n)$: + + \[ \Frobexp^n(-a) = -\Frobexp^n(a) = -a, \] + + \vskip 0.1in + + e che $a\inv \in \Fix(\Frobexp^n)$: + + \[ \Frobexp^n(a\inv) = \Frobexp^n(a)\inv = a\inv. \] + +\end{proof} + + +\subsection{Classificazione dei campi finiti} + +\begin{theorem} + Ogni campo finito $\KK$ di caratteristica $p$ consta + di $p^n$ elementi, con $n \in \NN^+$. +\end{theorem} + +\begin{proof} + Come già detto precedentemente, $\KK$ è uno + spazio vettoriale su una copia isomorfa di $\ZZp$, + $\FFpp$. \\ + + Si consideri allora il grado $[\KK : \FFpp]$. Sicuramente + questo grado non è infinito, dal momento che $\KK$ non + ha infiniti elementi. Quindi $[\KK : \FFpp] = n \in \NN$. \\ + + Sia dunque $(k_1, k_2, \ldots, k_n)$ una base di $\KK$ + su $\FFpp$. Ogni elemento $a$ di $\KK$ si potrà dunque scrivere + come: + + \[ a = \alpha_1 k_1 + \ldots + \alpha_n k_n, \quad \alpha_1, \ldots, \alpha_n \in \FFpp,\] + + \vskip 0.1in + + e dunque vi saranno in totale $p^n$ + elementi, dove ogni $p$ è contato dal numero di elementi che è + possibile associare ad ogni coefficiente, ossia $\card{\FFpp} = p$, + per il numero di elementi appartenenti alla base, ossia $[\KK : \FFpp] = + n$, da cui la tesi. +\end{proof} + +\begin{theorem} + Per ogni $n \in \NN^+$ e per ogni numero primo $p$ esiste un + campo finito con $p^n$ elementi. +\end{theorem} + +\begin{proof} + Si consideri il polinomio $x^{p^n}-x$ su $\ZZp$ e un suo + campo di spezzamento $A$. $\Fix(\Frobexp^n)$, per + la \propref{prop:punti_fissi_frobenius_campo}, è + un sottocampo, e + contiene esattamente le radici di $x^{p^n}-x$, che + in $A$ si spezza in fattori lineari, per definizione. \\ + + La derivata di $x^{p^n}-x$ è $p^n x^{p^n - 1}-1 \equiv -1$, + dacché $A$ è uno spazio vettoriale su $\ZZp$, e pertanto + vale ancora la \propref{prop:campo_char_p_prodotto_per_p}. + Dal momento che $-1$ e $x^{p^n}-x$ non hanno fattori lineari + in comune, per il \textit{Criterio della derivata}, + $x^{p^n}-x$ non ammette radici multiple. \\ + + Allora $\Fix(\Frobexp^n)$ è un campo con $p^n$ elementi, + ossia tutte le radici di $x^{p^n}-x$ (e coincide quindi + con il campo di spezzamento $A$), da cui la tesi. +\end{proof} diff --git a/Aritmetica/10. Teoremi rilevanti sui campi finiti.tex b/Aritmetica/11. Teoremi rilevanti sui campi finiti.tex similarity index 96% rename from Aritmetica/10. Teoremi rilevanti sui campi finiti.tex rename to Aritmetica/11. Teoremi rilevanti sui campi finiti.tex index 766ad74..07ca4ae 100644 --- a/Aritmetica/10. Teoremi rilevanti sui campi finiti.tex +++ b/Aritmetica/11. Teoremi rilevanti sui campi finiti.tex @@ -1,6 +1,6 @@ \section{Teoremi rilevanti sui campi finiti} -\subsection{Campo di spezzamento di un irriducibile in $\FFpp$} +\subsection{Campo di spezzamento di un irriducibile in \texorpdfstring{$\FFpp$}{F\_p}} \begin{theorem} Sia $f(x)$ un polinomio irriducibile in $\FFpp$ e sia @@ -97,7 +97,7 @@ \Lightning{}. \end{proof} -\subsection{L'inclusione $\FFpm \subseteq \FFpn$ e il polinomio $x^{p^n}-x$} +\subsection{L'inclusione \texorpdfstring{$\FFpm \subseteq \FFpn$}{F\_(p\string^m) in F\_(p\string^n)} e il polinomio \texorpdfstring{$x^{p^n}-x$}{x\string^(p\string^n)-x}} \begin{lemma} \label{lem:alpha_radice} diff --git a/Aritmetica/3. Esempi notevoli di anelli euclidei.tex b/Aritmetica/3. Esempi notevoli di anelli euclidei.tex index 3d528fc..a00a9d7 100644 --- a/Aritmetica/3. Esempi notevoli di anelli euclidei.tex +++ b/Aritmetica/3. Esempi notevoli di anelli euclidei.tex @@ -1,6 +1,6 @@ \section{Esempi notevoli di anelli euclidei} -\subsection{I numeri interi: $\ZZ$} +\subsection{I numeri interi: \texorpdfstring{$\ZZ$}{Z}} Senza ombra di dubbio l'esempio più importante di anello euclideo -- nonché l'esempio da cui si è generalizzata proprio la stessa nozione di anello @@ -20,7 +20,7 @@ Dal momento che così si verifica che $\ZZ$ è un anello euclideo, il \textit{Te fondamentale dell'aritmetica} è una conseguenza del \textit{Teorema \ref{th:euclidei_ufd}}. -\subsection{I campi: $\KK$} +\subsection{I campi: \texorpdfstring{$\KK$}{K}} Ogni campo $\KK$ è un anello euclideo, seppur banalmente. Infatti, eccetto proprio per $0$, ogni elemento è "divisibile" per ogni altro elemento: siano $a$, $b \in \KK$, @@ -36,7 +36,7 @@ Chiaramente $g$ soddisfa il primo assioma della funzione grado. Inoltre, poiché ogni elemento è "divisibile", il resto è sempre zero -- non è pertanto necessario verificare nessun'altra proprietà. -\subsection{I polinomi di un campo: $\KK[x]$} +\subsection{I polinomi di un campo: \texorpdfstring{$\KK[x]$}{K[x]}} I polinomi di un campo $\KK$ formano un anello euclideo rilevante nello studio dell'algebra astratta. Come suggerisce la @@ -62,7 +62,7 @@ euclideo\footnote{Curiosamente i polinomi di $\KK[x]$ e i campi $\KK$ sono gli u $\Ker \varphi = (x-\alpha)$. \end{example} -\subsection{Gli interi di Gauss: $\ZZ[i]$} +\subsection{Gli interi di Gauss: \texorpdfstring{$\ZZ[i]$}{Z[i]}} Un importante esempio di anello euclideo è il dominio degli interi di Gauss $\ZZ[i]$, definito come: @@ -109,8 +109,6 @@ di $\ZZ$ e quella di $\ZZ[i].$ Infatti, se $a \in \ZZ$, il grado di $a$ in $\ZZ$ sono uno il quadrato dell'altro. In particolare, è possibile ridefinire il grado di $\ZZ$ proprio in modo tale da farlo coincidere con quello di $\ZZ[i]$. \\ -\newpage - \begin{theorem} $\ZZ[i]$ è un anello euclideo. \end{theorem} @@ -138,7 +136,7 @@ di $\ZZ$ proprio in modo tale da farlo coincidere con quello di $\ZZ[i]$. \\ \[\left|r\right| \leq \frac{\left|b\right|}{\sqrt{2}} < \left|b\right| \implies \left|r\right|^2 < \left|b\right|^2 \implies g(r) < g(b).\] \end{proof} -\subsection{Gli interi di Eisenstein: $\ZZ[\omega]$} +\subsection{Gli interi di Eisenstein: \texorpdfstring{$\ZZ[\omega]$}{Z[ω]}} Sulla scia di $\ZZ[i]$ è possibile definire anche l'anello degli interi di Eisenstein, aggiungendo a $\ZZ$ la prima radice cubica diff --git a/Aritmetica/4. Proprietà fondamentali di Z[i], Zp[x], Z[x], Q[x].tex b/Aritmetica/4. Proprietà fondamentali di Z[i], Zp[x], Z[x], Q[x].tex index f06f0fb..d41386d 100644 --- a/Aritmetica/4. Proprietà fondamentali di Z[i], Zp[x], Z[x], Q[x].tex +++ b/Aritmetica/4. Proprietà fondamentali di Z[i], Zp[x], Z[x], Q[x].tex @@ -1,4 +1,4 @@ -\section{Irriducibili e corollari di aritmetica in $\ZZi$} +\section{Irriducibili e corollari di aritmetica in \texorpdfstring{$\ZZi$}{Z[i]}} Come già dimostrato, $\ZZi$ è un anello euclideo con la seguente funzione grado: @@ -10,7 +10,7 @@ importante in aritmetica, il \nameref{th:teorema_natale}, che discende direttamente come corollario di un teorema più generale riguardante $\ZZi$. -\subsection{Il teorema di Natale di Fermat e gli irriducibili in $\ZZi$} +\subsection{Il teorema di Natale di Fermat e gli irriducibili in \texorpdfstring{$\ZZi$}{Z[i]}} \begin{lemma} \label{lem:riducibile_due_quadrati} diff --git a/Aritmetica/5. Irriducibilità in Z[x] e Q[x].tex b/Aritmetica/5. Irriducibilità in Z[x] e Q[x].tex index 1394005..ae33b87 100644 --- a/Aritmetica/5. Irriducibilità in Z[x] e Q[x].tex +++ b/Aritmetica/5. Irriducibilità in Z[x] e Q[x].tex @@ -1,6 +1,6 @@ -\section{Irriducibilità in $\ZZx$ e in $\QQx$} +\section{Irriducibilità in \texorpdfstring{$\ZZx$}{Z[x]} e in \texorpdfstring{$\QQx$}{Q[x]}} -\subsection{Criterio di Eisenstein e proiezione in $\ZZpx$} +\subsection{Criterio di Eisenstein e proiezione in \texorpdfstring{$\ZZpx$}{Z\_p[x]}} Prima di studiare le irriducibilità in $\ZZ$, si guarda alle irriducibilità nei vari campi finiti $\ZZp$, con @@ -178,7 +178,7 @@ verrà ripresa anche in seguito irriducibile. Pertanto anche $f(x)$ lo è. \end{example} -\subsection{Alcuni irriducibili di $\ZZ_2[x]$} +\subsection{Alcuni irriducibili di \texorpdfstring{$\ZZ_2[x]$}{Z\_2[x]}} Tra tutti gli anelli $\ZZpx$, $\ZZ_2[x]$ ricopre sicuramente un ruolo fondamentale, dal momento che è il meno costoso diff --git a/Aritmetica/6. Proprietà dei polinomi di K[x] e delle estensioni algebriche.tex b/Aritmetica/6. Proprietà dei polinomi di K[x] e delle estensioni algebriche.tex index 6f87a2d..bc0e185 100644 --- a/Aritmetica/6. Proprietà dei polinomi di K[x] e delle estensioni algebriche.tex +++ b/Aritmetica/6. Proprietà dei polinomi di K[x] e delle estensioni algebriche.tex @@ -1,4 +1,4 @@ -\section{I polinomi di un campo: $\KKx$} +\section{I polinomi di un campo: \texorpdfstring{$\KKx$}{K[x]}} \subsection{Elementi preliminari} @@ -89,7 +89,7 @@ ora invece la definizione di radice. quindi un UFD, \Lightning{}. Quindi le radici sono esattamente $k \leq n$, da cui la tesi. \end{proof} -\subsection{Sottogruppi moltiplicativi finiti di $\KK$} +\subsection{Sottogruppi moltiplicativi finiti di \texorpdfstring{$\KK$}{K}} Si illustra adesso un teorema che riguarda i sottogruppi moltiplicativi finiti di $\KK$, da cui conseguirà, @@ -203,7 +203,7 @@ qualsiasi $p$ primo. \\ Quindi $\card{X_d}>0$, e $G$ è ciclico. \end{proof} -\subsection{Il quoziente $\KKx/(f(x))$} +\subsection{Il quoziente \texorpdfstring{$\KKx/(f(x))$}{K[x]/(f(x))}} Nell'ambito dello studio delle radici di un polinomio, il quoziente $\KKx/(f(x))$ gioca un ruolo fondamentale. diff --git a/Aritmetica/7. Estensioni algebriche di K.tex b/Aritmetica/7. Estensioni algebriche di K.tex index 9411ca1..4969aad 100644 --- a/Aritmetica/7. Estensioni algebriche di K.tex +++ b/Aritmetica/7. Estensioni algebriche di K.tex @@ -1,4 +1,4 @@ -\section{Estensioni algebriche di $\KK$} +\section{Estensioni algebriche di \texorpdfstring{$\KK$}{K}} \subsection{Morfismi di valutazione, elementi algebrici e trascendenti} diff --git a/Aritmetica/8. Campi di spezzamento.tex b/Aritmetica/8. Campi di spezzamento.tex index 55eb11e..5d26b17 100644 --- a/Aritmetica/8. Campi di spezzamento.tex +++ b/Aritmetica/8. Campi di spezzamento.tex @@ -50,4 +50,26 @@ Pertanto ora è possibile enunciare la definizione di \textit{campo di spezzamen In generale i campi di spezzamento non sono uguali, sebbene siano tutti isomorfi tra loro\footnote{Per la dimostrazione di questo risultato si rimanda a TODO}. -\end{remark*} \ No newline at end of file +\end{remark*} + +\begin{theorem} + Sia $A$ un campo e sia $B \supseteq A$ un campo di spezzamento + di $f(x) \in A[x]$ su $A$, con $f(x)$ non costante. Sia $\deg f(x) = n$. + Allora $[B : A] \leq n!$. +\end{theorem} + +\begin{proof} + Siano $\lambda_1$, $\lambda_2,\,\ldots,$ $\lambda_n$ le radici + di $f(x)$. Allora $[\KK(\lambda_1) : \KK] \leq n$, dacché + $\lambda_1$ è radice di $f(x)$. \\ + + Sia ora $f(x)=(x-\lambda_1)g(x)$, con $\deg g(x) = n-1$. Sicuramente + $\lambda_2$ è radice di $g(x)$, pertanto $[\KK(\lambda_1, \lambda_2) : \KK(\lambda_1)] \leq n-1$. Reiterando il ragionamento si può applicare infine il \nameref{th:torri}: + + \[ [\KK(\lambda_1, \ldots, \lambda_n) : \KK] = [\KK(\lambda_1, \ldots, \lambda_n) : \KK(\lambda_1, \ldots, \lambda_{n-1})] \cdots [\KK(\lambda_1) : \KK] \leq 1 \cdot 2 \cdots n = n!, \] + + \vskip 0.1in + + da cui la tesi. +\end{proof} + diff --git a/Aritmetica/9. Teorema fondamentale dell'algebra ed estensioni di Q.tex b/Aritmetica/9. Teorema fondamentale dell'algebra ed estensioni di Q.tex index f046f7b..25f5808 100644 --- a/Aritmetica/9. Teorema fondamentale dell'algebra ed estensioni di Q.tex +++ b/Aritmetica/9. Teorema fondamentale dell'algebra ed estensioni di Q.tex @@ -1,4 +1,4 @@ -\section{Teorema fondamentale dell'Algebra e radici reali in $\QQx$} +\section{Teorema fondamentale dell'Algebra e radici reali in \texorpdfstring{$\QQx$}{Q[]}} Si enuncia adesso il \nameref{th:algebra}, senza tuttavia fornirne una dimostrazione\footnote{Per la dimostrazione si rimanda diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index cc54c3c0c0e1dd708b084c59d5750872dda1991a..ba90f9726ab5e9c8a775660f949e798708513dad 100644 GIT binary patch delta 420532 zcmZ^L1yqz<_dY|nbazR^P(ydObf_jk(*K;OX%ZJl) zIOw~%tSWsl#@WjexZfE|44CP!zkdiXCG1gjV5tOZDh6$@cDjh%lZp|_{(ww{qv1k= zXaJQEMO`#>2tO^r7ZJ$KOU1hpf{A=piCX6n`kb95&4FE!G~Twd$~P3F<$93elJK{jqYFkL57G%CIV!d+x`&Uju3EJ7ll1D zd}G?~w(m;w@)P>nn>9+C)A4G2(x!f!0y%zpYFon~?9kf^jt4WO__tUtx|luGCC$2B z=l2ckrpwD_#XtwlrtYnx(G;|P{IZiko_?d&xTOqgqtBInR;>=q=!fUH+ zq_2)I7bfC<@~IpZvZiB;slt8kEdlDcUvkG1cXwP5$Q6{`G3Hw$&Shgq!p~p$VhLk+ 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zvx)|Ovn!@BT%QOje^%NSXMrbdw3p6vI<}#`Qi_FOBXBUfIq$%jlA5TH-eV+%v5GiMI;2Dl0du@Bq^1CkyqB!8j=uw=l(gR&?SV*AFt|fYM-rL5 z5z}vnTP(Q%k$-bT8I{6a1l3R;`coWS%BysWKd9`b{9N^=EJ=Im7zmw9a?wi%XwED_ ziAwJsuvX_8xejM)5~!{756ATuC!tAoWbqRD8aVcHkE<(Q@5`9*0 zlqfCcf>&%p{~3@v3a#zr!ibh|lET)(G5ijULp3p{Q>TZd-lChY-Xk0;jH#VvNo%_o ze-Ag2AekDs!Tqo>lV+|dY|`OxVB8o|ps~om?w-JJS|IWCnl=|3?mlWe z6_|<10yEDO029+N&KoT>F4|k>qTES)WO5vl2-JWpgjF&M}mSvi#2joD;N=(S`rfc*5&{=)moi{ zA@kEM844B}K$D&Vrt6vIN(7ZiY3~wQB~)i%*i;5)ia?rX%2_n~gtSiQGiH;h1>h*_ z0m*}#b|jfC%w!O_wMy=j@r3?!@?#5|m0(I+n0+D-h>`tb8S1U$Mnjj6{Ql#0Othzk zEo8EY!1|?FAP>xxsQ@!IGr&wv9@x|bHo3!rfu%@IIh=O!XlJT~fSI@jF!OK(ut-7H zKO;YtVn*B|MUB*>WZFnaDV@;#0V`;$Ufv(>jkm#*khsj)eb+a#JzP{CZ9H=4xpNof zU1}I)c<06=w>Iy`Bo5`ug$s=9_Q2;fCw3zy5}H ra&uiA8~*>BoEjcGw(+qGr_Wz_;{2HhHiBEbp=mcBIB?U6n>YSHyu|bt diff --git a/Aritmetica/aritmetica.tex b/Aritmetica/aritmetica.tex index 5c9f883..680a295 100644 --- a/Aritmetica/aritmetica.tex +++ b/Aritmetica/aritmetica.tex @@ -1,6 +1,6 @@ \PassOptionsToPackage{main=italian}{babel} \documentclass[11pt]{scrartcl} -\usepackage[sexy]{evan} +\usepackage{evan} \usepackage[utf8]{inputenc} \usepackage[italian]{babel} \usepackage{algorithm2e} @@ -19,96 +19,6 @@ \addbibresource{bibliography.bib} -\newcommand{\nsqrt}[2]{\!\sqrt[#1]{#2}\,} -\newcommand{\card}[1]{\left|#1\right|} - -\newcommand{\zeroset}{\{0\}} -\newcommand{\setminuszero}{\setminus \{0\}} - -\newcommand{\corref}[1]{\textit{Corollario \ref{#1}}} -\newcommand{\exref}[1]{\textit{Esercizio \ref{#1}}} -\newcommand{\exmplref}[1]{\textit{Esempio \ref{#1}}} -\newcommand{\propref}[1]{\textit{Proposizione \ref{#1}}} -\newcommand{\lemref}[1]{\textit{Lemma \ref{#1}}} -\newcommand{\thref}[1]{\textit{Teorema \ref{#1}}} - -\newcommand{\BB}{\mathcal{B}} -\newcommand{\HH}{\mathbb{H}} -\newcommand{\KK}{\mathbb{K}} -\newcommand{\ZZp}{\mathbb{Z}_p} - -\newcommand{\CCx}{\mathbb{C}[x]} -\newcommand{\FFpp}{\mathbb{F}_p} -\newcommand{\FFpd}{\mathbb{F}_{p^d}} -\newcommand{\FFpm}{\mathbb{F}_{p^m}} -\newcommand{\FFpn}{\mathbb{F}_{p^n}} -\newcommand{\FFp}[1]{\mathbb{F}_{p^{#1}}} -\newcommand{\KKx}{\mathbb{K}[x]} -\newcommand{\QQx}{\mathbb{Q}[x]} -\newcommand{\RRx}{\mathbb{R}[x]} -\newcommand{\ZZi}{\mathbb{Z}[i]} -\newcommand{\ZZom}{\mathbb{Z}[\omega]} -\newcommand{\ZZpx}{\mathbb{Z}_p[x]} -\newcommand{\ZZx}{\mathbb{Z}[x]} - -\renewcommand{\ii}{\mathbf{i}} -\newcommand{\jj}{\mathbf{j}} -\newcommand{\kk}{\mathbf{k}} - -\newcommand{\valalpha}{\varphi_\alpha} -\newcommand{\Frob}{\mathcal{F}} -\newcommand{\Frobexp}{\mathcal{F}{\mkern 1.5mu}} - -\newcommand{\dual}[1]{#1^{*}} -\newcommand{\LL}[2]{\mathcal{L} \left(#1, \, #2\right)} -\newcommand{\M}[1]{\mathcal{M}_{#1}\left(\KK\right)} -\newcommand{\nsg}{\mathrel{\unlhd}} -\renewcommand{\vec}[1]{\underline{#1}} - -\newcommand{\hatpi}{\hat{\pi}} -\newcommand{\hatpip}{\hat{\pi}_p} - -\theoremstyle{definition} - -\DeclareMathOperator{\existsone}{\exists !} -\DeclareMathOperator{\Imm}{Imm} -\DeclareMathOperator{\MCD}{MCD} -\DeclareMathOperator{\mcm}{mcm} -\DeclareMathOperator{\tr}{tr} - -\let\oldemptyset\emptyset -\let\emptyset\varnothing - -\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} - -\setlength\parindent{0pt} - \begin{document} \title{L'Algebrario} @@ -192,7 +102,13 @@ \thispagestyle{empty} ~\newpage -\include{10. Teoremi rilevanti sui campi finiti} +\include{10. Introduzione a teoria dei campi} + +\newpage +\thispagestyle{empty} +~\newpage + +\include{11. Teoremi rilevanti sui campi finiti} \newpage \thispagestyle{empty} diff --git a/Aritmetica/bibliography.bib b/Aritmetica/bibliography.bib index 7066734..f5e54ac 100644 --- a/Aritmetica/bibliography.bib +++ b/Aritmetica/bibliography.bib @@ -6,7 +6,7 @@ year={2013}, publisher={Pisa University Press}, shorthand={DM} -}, +} @book{herstein2010algebra, title={Algebra}, author={Herstein, I.N.}, @@ -14,28 +14,28 @@ year={2010}, publisher={Editori Riuniti University Press}, shorthand={H} -}, +} @Inbook{Remmert1991, -author="Remmert, R.", -title="The Fundamental Theorem of Algebra", -bookTitle="Numbers", -year="1991", -publisher="Springer New York", -address="New York, NY", -pages="97--122", -isbn="978-1-4612-1005-4", -doi="10.1007/978-1-4612-1005-4_5", -url="https://doi.org/10.1007/978-1-4612-1005-4_5" -}, + author="Remmert, R.", + title="The Fundamental Theorem of Algebra", + bookTitle="Numbers", + year="1991", + publisher="Springer New York", + address="New York, NY", + pages="97--122", + isbn="978-1-4612-1005-4", + doi="10.1007/978-1-4612-1005-4_5", + url="https://doi.org/10.1007/978-1-4612-1005-4_5" +} @article{10.2307/2315810, - ISSN = {00029890, 19300972}, - URL = {http://www.jstor.org/stable/2315810}, - author = {M. A. Jodeit}, - journal = {The American Mathematical Monthly}, - number = {7}, - pages = {835--836}, - publisher = {Mathematical Association of America}, - title = {Uniqueness in the Division Algorithm}, - volume = {74}, - year = {1967} + ISSN = {00029890, 19300972}, + URL = {http://www.jstor.org/stable/2315810}, + author = {M. A. Jodeit}, + journal = {The American Mathematical Monthly}, + number = {7}, + pages = {835--836}, + publisher = {Mathematical Association of America}, + title = {Uniqueness in the Division Algorithm}, + volume = {74}, + year = {1967} } \ No newline at end of file diff --git a/Aritmetica/evan.sty b/Aritmetica/evan.sty index 42f666d..621b0af 100644 --- a/Aritmetica/evan.sty +++ b/Aritmetica/evan.sty @@ -12,16 +12,9 @@ % \▓▓▓▓▓▓ % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Original file: % https://github.com/vEnhance/dotfiles/blob/main/texmf/tex/latex/evan/evan.sty -% ░█▄█░▄▀▄░█░░▒█░░░▀█▀░▄▀▄░░░█▒█░▄▀▀▒██▀ -% ▒█▒█░▀▄▀░▀▄▀▄▀▒░░▒█▒░▀▄▀▒░░▀▄█▒▄██░█▄▄ -% -% If you don't know how to use this file, read: -% +--------------------------------------------+ -% | https://web.evanchen.cc/faq-latex.html#L-4 | -% +--------------------------------------------+ -% % TL;DR of the Boost license conditions are as follows: % % 1. Any SOURCE VERSIONS must cite evan.sty and the Boost license below. @@ -62,145 +55,88 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \ProvidesPackage{evan} -%%fakesection Argument processing -% Default Arguments -% We include "Evan" in all of these to make sure -% that they don't collide with anything in external packages -\newif\ifevanfancy\evanfancytrue -\newif\ifevanhdr\evanhdrtrue -\newif\ifevanhref\evanhreftrue -\newif\ifevansetup\evansetuptrue -\newif\ifevanthm\evanthmtrue -\newif\ifevansecthm\evansecthmfalse -\newif\ifevanht\evanhtfalse -\newif\ifevanpkg\evanpkgtrue -\newif\ifevanpdf\evanpdftrue -\newif\ifevanauthor\evanauthortrue -\newif\ifevanchinese\evanchinesefalse -\newif\ifevanmdthm\evanmdthmfalse -\newif\ifevandiagrams\evandiagramsfalse -\newif\ifevanpatchasy\evanpatchasyfalse -\newif\ifevanhints\evanhintsfalse -\newif\ifevanasy\evanasytrue -\newif\ifevancolorsec\evancolorsecfalse -\newif\ifevantitlemark\evantitlemarktrue -\newif\ifevanvonenabled\evanvonenabledfalse -\newif\ifevanbritish\evanbritishtrue - -%Receive Arguments -\DeclareOption{chinese}{\evanhreffalse\evanchinesetrue} % Chinese support -% allow href to override this one - -\DeclareOption{sexy}{\evansecthmtrue\evanmdthmtrue\evancolorsectrue} % long docs - -\DeclareOption{fancy}{\evanfancytrue} -\DeclareOption{nofancy}{\evanfancyfalse} -\DeclareOption{hdr}{\evanhdrtrue} -\DeclareOption{nohdr}{\evanhdrfalse} -\DeclareOption{href}{\evanhreftrue} -\DeclareOption{nohref}{\evanhreffalse} - -\DeclareOption{nosetup}{\evansetupfalse} -\DeclareOption{thm}{\evanthmtrue} -\DeclareOption{nothm}{\evanthmfalse} -\DeclareOption{secthm}{\evansecthmtrue} -\DeclareOption{nosecthm}{\evansecthmfalse} - -\DeclareOption{ht}{\evanhttrue} -\DeclareOption{nopdf}{\evanpdffalse} -\DeclareOption{nopkg}{\evanpkgfalse} -\DeclareOption{noauthor}{\evanauthorfalse} -\DeclareOption{titlemark}{\evantitlemarktrue} % Sets title in ohead, not \rightmark -\DeclareOption{sectionmark}{\evantitlemarkfalse} % Uses \rightmark not title in ohead - -\DeclareOption{mdthm}{\evanmdthmtrue} -\DeclareOption{nomdthm}{\evanmdthmfalse} -\DeclareOption{diagrams}{\evandiagramstrue} -\DeclareOption{nodiagrams}{\evandiagramsfalse} -\DeclareOption{colorsec}{\evancolorsectrue} -\DeclareOption{nocolorsec}{\evancolorsecfalse} - -\DeclareOption{patchasy}{\evanpatchasytrue} -\DeclareOption{noasy}{\evanasyfalse} - -\DeclareOption{hints}{\evanhintstrue} -\DeclareOption{von}{\evanvonenabledtrue} - -\DeclareOption{british}{\evanbritishtrue} -\DeclareOption{american}{\evanbritishfalse} \ProcessOptions\relax -% if packages not loaded, turn off mdthm and asy -\ifevanpkg\else\evanmdthmfalse\fi -\ifevanpkg\else\evanasyfalse\fi - -% If no setup, turn off theorems -\ifevansetup\else\evanthmfalse\fi - -%%fakesection Some macros -%Small commands \usepackage{amsmath,amssymb} \usepackage{iftex} -\ifevanpkg - \usepackage[minimal]{yhmath} -\fi +\usepackage[minimal]{yhmath} + +\setlength\parindent{0pt} + +% Personal commands +\newcommand{\card}[1]{\left|#1\right|} +\newcommand{\nsqrt}[2]{\!\sqrt[#1]{#2}\,} +\newcommand{\zeroset}{\{0\}} +\newcommand{\setminuszero}{\setminus \{0\}} + +\newcommand{\corref}[1]{\textit{Corollario \ref{#1}}} +\newcommand{\exref}[1]{\textit{Esercizio \ref{#1}}} +\newcommand{\exmplref}[1]{\textit{Esempio \ref{#1}}} +\newcommand{\propref}[1]{\textit{Proposizione \ref{#1}}} +\newcommand{\lemref}[1]{\textit{Lemma \ref{#1}}} +\newcommand{\thref}[1]{\textit{Teorema \ref{#1}}} + +\newcommand{\BB}{\mathcal{B}} +\newcommand{\HH}{\mathbb{H}} +\newcommand{\KK}{\mathbb{K}} +\newcommand{\ZZp}{\mathbb{Z}_p} + +\newcommand{\CCx}{\mathbb{C}[x]} +\newcommand{\FFpp}{\mathbb{F}_p} +\newcommand{\FFpd}{\mathbb{F}_{p^d}} +\newcommand{\FFpm}{\mathbb{F}_{p^m}} +\newcommand{\FFpn}{\mathbb{F}_{p^n}} +\newcommand{\FFp}[1]{\mathbb{F}_{p^{#1}}} +\newcommand{\KKx}{\mathbb{K}[x]} +\newcommand{\QQx}{\mathbb{Q}[x]} +\newcommand{\RRx}{\mathbb{R}[x]} +\newcommand{\ZZi}{\mathbb{Z}[i]} +\newcommand{\ZZom}{\mathbb{Z}[\omega]} +\newcommand{\ZZpx}{\mathbb{Z}_p[x]} +\newcommand{\ZZx}{\mathbb{Z}[x]} + +\newcommand{\ii}{\mathbf{i}} +\newcommand{\jj}{\mathbf{j}} +\newcommand{\kk}{\mathbf{k}} + +\newcommand{\valalpha}{\varphi_\alpha} +\newcommand{\Frob}{\mathcal{F}} +\newcommand{\Frobexp}{\mathcal{F}{\mkern 1.5mu}} + +\newcommand{\dual}[1]{#1^{*}} +\newcommand{\LL}[2]{\mathcal{L} \left(#1, \, #2\right)} +\newcommand{\M}[1]{\mathcal{M}_{#1}\left(\KK\right)} +\newcommand{\nsg}{\mathrel{\unlhd}} +\renewcommand{\vec}[1]{\underline{#1}} + +\newcommand{\hatpi}{\hat{\pi}} +\newcommand{\hatpip}{\hat{\pi}_p} + +% 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{\ol}{\overline} -\newcommand{\ul}{\underline} -\newcommand{\wt}{\widetilde} -\newcommand{\wh}{\widehat} \newcommand{\eps}{\varepsilon} -%\renewcommand{\iff}{\Leftrightarrow} -%\renewcommand{\implies}{\Rightarrow} \newcommand{\vocab}[1]{\textbf{\color{blue}\sffamily #1}} \providecommand{\alert}{\vocab} -\providecommand{\half}{\frac{1}{2}} \newcommand{\catname}{\mathsf} -\newcommand{\hrulebar}{ - \par\hspace{\fill}\rule{0.95\linewidth}{.7pt}\hspace{\fill} - \par\nointerlineskip \vspace{\baselineskip} -} \providecommand{\arc}[1]{\wideparen{#1}} -%For use in author command -\newcommand{\plusemail}[1]{\\ \normalfont \texttt{\mailto{#1}}} - -%More commands and math operators -\DeclareMathOperator{\cis}{cis} -\DeclareMathOperator*{\lcm}{lcm} -\DeclareMathOperator*{\argmin}{arg min} -\DeclareMathOperator*{\argmax}{arg max} - -%Convenient Environments -\newenvironment{soln}{\begin{proof}[Solution]}{\end{proof}} -\newenvironment{parlist}{\begin{inparaenum}[(i)]}{\end{inparaenum}} -\newenvironment{gobble}{\setbox\z@\vbox\bgroup}{\egroup} - -%Inequalities -\newcommand{\cycsum}{\sum_{\mathrm{cyc}}} -\newcommand{\symsum}{\sum_{\mathrm{sym}}} -\newcommand{\cycprod}{\prod_{\mathrm{cyc}}} -\newcommand{\symprod}{\prod_{\mathrm{sym}}} - -%From H113 "Introduction to Abstract Algebra" at UC Berkeley +% 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} -\newcommand{\charin}{\text{ char }} -\DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Inn}{Inn} \DeclareMathOperator{\Syl}{Syl} \DeclareMathOperator{\Gal}{Gal} -\DeclareMathOperator{\GL}{GL} % General linear group -\DeclareMathOperator{\SL}{SL} % Special linear group +\DeclareMathOperator{\GL}{GL} +\DeclareMathOperator{\SL}{SL} %From Kiran Kedlaya's "Geometry Unbound" \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} @@ -209,619 +145,328 @@ \newcommand{\ray}[1]{\overrightarrow{#1}} \newcommand{\seg}[1]{\overline{#1}} -%From M275 "Topology" at SJSU +% From M275 "Topology" at SJSU \newcommand{\id}{\mathrm{id}} \newcommand{\taking}[1]{\xrightarrow{#1}} \newcommand{\inv}{^{-1}} -%From M170 "Introduction to Graph Theory" at SJSU -\DeclareMathOperator{\diam}{diam} \DeclareMathOperator{\ord}{ord} \newcommand{\defeq}{\overset{\mathrm{def}}{=}} -%From the USAMO .tex files -\newcommand{\ts}{\textsuperscript} +% From the USAMO .tex files \newcommand{\dg}{^\circ} -\newcommand{\ii}{\item} - -% From Math 55 and Math 145 at Harvard -\newenvironment{subproof}[1][Proof]{% -\begin{proof}[#1] \renewcommand{\qedsymbol}{$\blacksquare$}}% -{\end{proof}} \newcommand{\liff}{\leftrightarrow} \newcommand{\lthen}{\rightarrow} \newcommand{\opname}{\operatorname} \newcommand{\surjto}{\twoheadrightarrow} \newcommand{\injto}{\hookrightarrow} -\newcommand{\On}{\mathrm{On}} % ordinals -\DeclareMathOperator{\img}{im} % Image -\DeclareMathOperator{\Img}{Im} % Image -\DeclareMathOperator{\coker}{coker} % Cokernel -\DeclareMathOperator{\Coker}{Coker} % Cokernel -\DeclareMathOperator{\Ker}{Ker} % Kernel +\DeclareMathOperator{\Char}{char} +\DeclareMathOperator{\Dom}{Dom} +\DeclareMathOperator{\Fix}{\textit{Fix}\,} +\DeclareMathOperator{\End}{End} +\DeclareMathOperator{\existsone}{\exists !} +\DeclareMathOperator{\Hom}{Hom} +\DeclareMathOperator{\Imm}{Imm} +\DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\rank}{rank} -\DeclareMathOperator{\Spec}{Spec} % spectrum -\DeclareMathOperator{\Tr}{Tr} % trace -\DeclareMathOperator{\pr}{pr} % projection -\DeclareMathOperator{\ext}{ext} % extension -\DeclareMathOperator{\pred}{pred} % predecessor -\DeclareMathOperator{\dom}{dom} % domain -\DeclareMathOperator{\ran}{ran} % range -\DeclareMathOperator{\Hom}{Hom} % homomorphism -\DeclareMathOperator{\Mor}{Mor} % morphisms -\DeclareMathOperator{\End}{End} % endomorphism - -% Things Lie -\newcommand{\kb}{\mathfrak b} -\newcommand{\kg}{\mathfrak g} -\newcommand{\kh}{\mathfrak h} -\newcommand{\kn}{\mathfrak n} -\newcommand{\ku}{\mathfrak u} -\newcommand{\kz}{\mathfrak z} -\DeclareMathOperator{\Ext}{Ext} % Ext functor -\DeclareMathOperator{\Tor}{Tor} % Tor functor -\newcommand{\gl}{\opname{\mathfrak{gl}}} % frak gl group -\def\sl{\opname{\mathfrak{sl}}} % frak sl group chktex 6 - -% More script letters etc. -\newcommand{\SA}{\mathcal A} -\newcommand{\SB}{\mathcal B} -\newcommand{\SC}{\mathcal C} -\newcommand{\SF}{\mathcal F} -\newcommand{\SG}{\mathcal G} -\newcommand{\SH}{\mathcal H} -\newcommand{\OO}{\mathcal O} - -\newcommand{\SCA}{\mathscr A} -\newcommand{\SCB}{\mathscr B} -\newcommand{\SCC}{\mathscr C} -\newcommand{\SCD}{\mathscr D} -\newcommand{\SCE}{\mathscr E} -\newcommand{\SCF}{\mathscr F} -\newcommand{\SCG}{\mathscr G} -\newcommand{\SCH}{\mathscr H} - -% Mathfrak primes -\newcommand{\km}{\mathfrak m} -\newcommand{\kp}{\mathfrak p} -\newcommand{\kq}{\mathfrak q} - -%% Napkin commands -\newcommand{\prototype}[1]{ - \emph{{\color{red} Prototypical example for this section:} #1} \par\medskip +\DeclareMathOperator{\MCD}{MCD} +\DeclareMathOperator{\Mor}{Mor} +\DeclareMathOperator{\mcm}{mcm} +\DeclareMathOperator{\tr}{tr} + +\let\oldemptyset\emptyset +\let\emptyset\varnothing + +\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} + +\title{} +\author{Gabriel Antonio Videtta} +\date{\today} + +\PassOptionsToPackage{usenames,svgnames,dvipsnames,table}{xcolor} +\usepackage{xcolor} +\usepackage[colorlinks=true]{hyperref} +\hypersetup{urlcolor=RubineRed,linkcolor=RoyalBlue,citecolor=ForestGreen} +\usepackage[nameinlink]{cleveref} + +\usepackage{amsthm} +\usepackage{thmtools} +\usepackage[framemethod=TikZ]{mdframed} + +\mdfdefinestyle{mdbluebox}{% + roundcorner=10pt, + linewidth=1pt, + skipabove=12pt, + innerbottommargin=9pt, + skipbelow=2pt, + linecolor=blue, + nobreak=true, + backgroundcolor=TealBlue!5, } -\newenvironment{moral}{% - \begin{mdframed}[linecolor=green!70!black]% - \bfseries\color{green!70!black}}% - {\end{mdframed}} - -%%fakesection Asymptote setup -\ifevanasy - \ifevanpatchasy - \usepackage{patchasy} - \else - \usepackage{asymptote} - \fi - \begin{asydef} - defaultpen(fontsize(10pt)); - size(8cm); // set a reasonable default - usepackage("amsmath"); - usepackage("amssymb"); - settings.tex="pdflatex"; - settings.outformat="pdf"; - // Replacement for olympiad+cse5 which is not standard - import geometry; - // recalibrate fill and filldraw for conics - void filldraw(picture pic = currentpicture, conic g, pen fillpen=defaultpen, pen drawpen=defaultpen) - { filldraw(pic, (path) g, fillpen, drawpen); } - void fill(picture pic = currentpicture, conic g, pen p=defaultpen) - { filldraw(pic, (path) g, p); } - // some geometry - pair foot(pair P, pair A, pair B) { return foot(triangle(A,B,P).VC); } - pair orthocenter(pair A, pair B, pair C) { return orthocentercenter(A,B,C); } - pair centroid(pair A, pair B, pair C) { return (A+B+C)/3; } - // cse5 abbreviations - path CP(pair P, pair A) { return circle(P, abs(A-P)); } - path CR(pair P, real r) { return circle(P, r); } - pair IP(path p, path q) { return intersectionpoints(p,q)[0]; } - pair OP(path p, path q) { return intersectionpoints(p,q)[1]; } - path Line(pair A, pair B, real a=0.6, real b=a) { return (a*(A-B)+A)--(b*(B-A)+B); } - // cse5 more useful functions - picture CC() { - picture p=rotate(0)*currentpicture; - currentpicture.erase(); - return p; - } - pair MP(Label s, pair A, pair B = plain.S, pen p = defaultpen) { - Label L = s; - L.s = "$"+s.s+"$"; - label(L, A, B, p); - return A; - } - pair Drawing(Label s = "", pair A, pair B = plain.S, pen p = defaultpen) { - dot(MP(s, A, B, p), p); - return A; - } - path Drawing(path g, pen p = defaultpen, arrowbar ar = None) { - draw(g, p, ar); - return g; - } - \end{asydef} -\fi - -%%fakesection BEGIN MAIN SETUP -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\ifevansetup - %%fakesection Set up author and date - \ifevanauthor - \title{} % empty title to avoid crashes - \ifluatex - \author{Gabriel Antonio Videtta} - \else - \author{Gabriel Antonio Videtta} - \fi - \date{\today} - \fi - %%fakesection Hyperref - \ifevanpkg - \PassOptionsToPackage{usenames,svgnames,dvipsnames,table}{xcolor} - \usepackage{xcolor} - \ifevanhref - \usepackage[colorlinks=true]{hyperref} - \hypersetup{urlcolor=RubineRed,linkcolor=RoyalBlue,citecolor=ForestGreen} - \fi - \usepackage[nameinlink]{cleveref} - \fi - - %%fakesection New theorem styles - \ifevanthm - \usepackage{amsthm} - \usepackage{thmtools} - \fi - \ifevanmdthm - \ifevanthm - %% theorem packages loaded already - \else - \usepackage{amsthm} - \usepackage{thmtools} - \fi - \usepackage[framemethod=TikZ]{mdframed} - - \mdfdefinestyle{mdbluebox}{% - roundcorner=10pt, - linewidth=1pt, - skipabove=12pt, - innerbottommargin=9pt, - skipbelow=2pt, - linecolor=blue, - nobreak=true, - backgroundcolor=TealBlue!5, - } - \declaretheoremstyle[ - headfont=\sffamily\bfseries\color{MidnightBlue}, - mdframed={style=mdbluebox}, - headpunct={\\[3pt]}, - postheadspace={0pt} - ]{thmbluebox} - - \mdfdefinestyle{mdbluebox2}{% - roundcorner=10pt, - linewidth=1pt, - skipabove=12pt, - innerbottommargin=9pt, - skipbelow=2pt, - linecolor=blue, - nobreak=true, - backgroundcolor=BlueViolet!9, - } - \declaretheoremstyle[ - headfont=\sffamily\bfseries\color{RoyalPurple}, - mdframed={style=mdbluebox2}, - headpunct={\\[3pt]}, - postheadspace={0pt} - ]{thmbluebox2} - - \mdfdefinestyle{mdredbox}{% - linewidth=0.5pt, - skipabove=12pt, - frametitleaboveskip=5pt, - frametitlebelowskip=0pt, - skipbelow=2pt, - frametitlefont=\bfseries, - innertopmargin=4pt, - innerbottommargin=8pt, - nobreak=true, - backgroundcolor=Salmon!5, - linecolor=RawSienna, - } - \declaretheoremstyle[ - headfont=\bfseries\color{RawSienna}, - mdframed={style=mdredbox}, - headpunct={\\[3pt]}, - postheadspace={0pt}, - ]{thmredbox} - - \mdfdefinestyle{mdredbox2}{% - roundcorner=10pt, - linewidth=1pt, - skipabove=12pt, - innerbottommargin=9pt, - skipbelow=2pt, - linecolor=red, - nobreak=true, - backgroundcolor=WildStrawberry!5, - } - \declaretheoremstyle[ - headfont=\sffamily\bfseries\color{Maroon}, - mdframed={style=mdredbox2}, - headpunct={\\[3pt]}, - postheadspace={0pt} - ]{thmredbox2} - - \mdfdefinestyle{mdgreenbox}{% - skipabove=8pt, - linewidth=2pt, - rightline=false, - leftline=true, - topline=false, - bottomline=false, - linecolor=ForestGreen, - backgroundcolor=ForestGreen!5, - } - \declaretheoremstyle[ - headfont=\bfseries\sffamily\color{ForestGreen!70!black}, - bodyfont=\normalfont, - spaceabove=2pt, - spacebelow=1pt, - mdframed={style=mdgreenbox}, - headpunct={ --- }, - ]{thmgreenbox} - - \mdfdefinestyle{mdgreenbox2}{% - roundcorner=10pt, - linewidth=1pt, - skipabove=12pt, - innerbottommargin=9pt, - skipbelow=2pt, - linecolor=ForestGreen, - nobreak=true, - backgroundcolor=ForestGreen!5, - } - \declaretheoremstyle[ - headfont=\sffamily\bfseries\color{ForestGreen!70!black}, - mdframed={style=mdgreenbox2}, - headpunct={\\[3pt]}, - postheadspace={0pt} - ]{thmgreenbox2} - - \mdfdefinestyle{mdblackbox}{% - skipabove=8pt, - linewidth=3pt, - rightline=false, - leftline=true, - topline=false, - bottomline=false, - linecolor=black, - backgroundcolor=RedViolet!5!gray!10, - } - \declaretheoremstyle[ - headfont=\bfseries, - bodyfont=\normalfont\small, - spaceabove=0pt, - spacebelow=0pt, - mdframed={style=mdblackbox} - ]{thmblackbox} - - \mdfdefinestyle{mdblackbox2}{% - skipabove=8pt, - linewidth=3pt, - rightline=false, - leftline=true, - topline=false, - bottomline=false, - linecolor=gray, - backgroundcolor=RedViolet!5!gray!10, - } - \declaretheoremstyle[ - headfont=\bfseries, - bodyfont=\normalfont\small, - spaceabove=0pt, - spacebelow=0pt, - mdframed={style=mdblackbox2} - ]{thmblackbox2} - - \newcommand{\listhack}{$\empty$\vspace{-2em}} - \fi - - %%fakesection Theorem setup - \ifevanthm - \theoremstyle{definition} - %Branching here: the option secthm changes theorems to be labelled by section - \ifevanmdthm - \ifevansecthm - \declaretheorem[% - style=thmbluebox,name=Teorema,numberwithin=section]{theorem} - \else - \declaretheorem[% - style=thmbluebox,name=Teorema]{theorem} - \fi - \declaretheorem[style=thmgreenbox2,name=Lemma,sibling=theorem]{lemma} - \declaretheorem[style=thmredbox2,name=Proposizione,sibling=theorem]{proposition} - \declaretheorem[style=thmbluebox2,name=Corollario,sibling=theorem]{corollary} - \declaretheorem[style=thmbluebox,name=Teorema,numbered=no]{theorem*} - \declaretheorem[style=thmgreenbox2,name=Lemma,numbered=no]{lemma*} - \declaretheorem[style=thmredbox2,name=Proposizione,numbered=no]{proposition*} - \declaretheorem[style=thmbluebox2,name=Corollario,numbered=no]{corollary*} - \else - \ifevansecthm - \declaretheorem[name=Teorema,numberwithin=section]{theorem} - \else - \declaretheorem[name=Teorema]{theorem} - \fi - \declaretheorem[name=Lemma,sibling=theorem]{lemma} - \declaretheorem[name=Proposizione,sibling=theorem]{proposition} - \declaretheorem[name=Corollario,sibling=theorem]{corollary} - \declaretheorem[name=Teorema,numbered=no]{theorem*} - \declaretheorem[name=Lemma,numbered=no]{lemma*} - \declaretheorem[name=Proposizione,numbered=no]{proposition*} - \declaretheorem[name=Corollario,numbered=no]{corollary*} - \fi - - \ifevanmdthm - \declaretheorem[style=thmgreenbox,name=Claim,sibling=theorem]{claim} - \declaretheorem[style=thmgreenbox,name=Claim,numbered=no]{claim*} - \else - \declaretheorem[name=Claim,sibling=theorem]{claim} - \declaretheorem[name=Claim,numbered=no]{claim*} - \fi - - \ifevanmdthm - \declaretheorem[style=thmredbox,name=Esempio,sibling=theorem]{example} - \declaretheorem[style=thmredbox,name=Esempio,numbered=no]{example*} - \else - \declaretheorem[name=Esempio,sibling=theorem]{example} - \declaretheorem[name=Esempio,numbered=no]{example*} - \fi - - % Remark-style theorems - %\theoremstyle{remark} - \ifevanmdthm - \declaretheorem[style=thmblackbox, name=Definizione,sibling=theorem]{definition} - \declaretheorem[style=thmblackbox, name=Definizione,numbered=no]{definition*} - \declaretheorem[style=thmblackbox2,name=Osservazione,sibling=theorem]{remark} - \declaretheorem[style=thmblackbox2,name=Osservazione,numbered=no]{remark*} - \else - \declaretheorem[name=Definizione,sibling=theorem]{definition} - \declaretheorem[name=Definizione,numbered=no]{definition*} - \declaretheorem[name=Osservazione,sibling=theorem]{remark} - \declaretheorem[name=Osservazione,numbered=no]{remark*} - \fi - - \declaretheorem[name=Congettura,sibling=theorem]{conjecture} - \declaretheorem[name=Congettura,numbered=no]{conjecture*} - \declaretheorem[name=Esercizio,sibling=theorem]{exercise} - \declaretheorem[name=Esercizio,numbered=no]{exercise*} - \declaretheorem[name=Asserzione,sibling=theorem]{fact} - \declaretheorem[name=Asserzione,numbered=no]{fact*} - \declaretheorem[name=Problema,sibling=theorem]{problem} - \declaretheorem[name=Problema,numbered=no]{problem*} - \declaretheorem[name=Domanda,sibling=theorem]{ques} - \declaretheorem[name=Domanda,numbered=no]{ques*} - - \ifevanpkg - \Crefname{claim}{Claim}{Claim} - \Crefname{conjecture}{Congettura}{Congetture} - \Crefname{exercise}{Esercizio}{Esercizi} - \Crefname{fact}{Asserzione}{Asserzioni} - \Crefname{problem}{Problema}{Problemi} - \Crefname{ques}{Domanda}{Domande} - \fi - \fi - - %%fakesection Fancy section and chapter heads - \ifevancolorsec - \@ifundefined{KOMAClassName}{}{ - \@ifundefined{chapter}{}{ - \addtokomafont{partprefix}{\rmfamily} - \renewcommand*{\partformat}{\color{purple} - \scalebox{2.5}{\thepart}\enlargethispage{2em}} - \addtokomafont{chapterprefix}{\raggedleft} - \RedeclareSectionCommand[beforeskip=0.5em]{chapter} - \renewcommand*{\chapterformat}{\mbox{% - \scalebox{1.5}{\chapappifchapterprefix{\nobreakspace}}% - \scalebox{2.718}{\color{purple}\thechapter}\enskip}} - } - \renewcommand*{\sectionformat}% - {\color{purple}\S\thesection\enskip} - \renewcommand*{\subsectionformat}% - {\color{purple}\S\thesubsection\enskip} - \renewcommand*{\subsubsectionformat}% - {\color{purple}\S\thesubsubsection\enskip} - \KOMAoptions{numbers=noenddot} - %\usetocstyle{KOMAlike} - } - \fi - - - %%fakesection Loads a bunch of useful packages (but allow disabling) - \ifevanpkg - \ifevanvonenabled - \IfFileExists{von.sty}{\usepackage{von}}{} - \fi - \usepackage{listings} - \usepackage{mathrsfs} - \usepackage{textcomp} - \lstset{basicstyle=\ttfamily\scriptsize, - backgroundcolor=\color{green!2!white}, - breakatwhitespace=true, - breaklines=true, - commentstyle=\color{green!70!black}, - frame=shadowbox, - frame=single, - identifierstyle=\color{green!20!black}, - keywordstyle=\bfseries, - keywordstyle=\bfseries\color{blue}, - numbers=left, - numbersep=5pt, - numberstyle=\tiny\sffamily\itshape\color{black!60}, - rulecolor=\color{blue!70!black}, - rulesepcolor=\color{blue!30!black}, - showstringspaces=false, - stringstyle=\color{orange}, - tabsize=4, - } % chktex 6 - \lstdefinelanguage{gitcommit}{ - alsoletter={:}, - morecomment=[l]{|}, - morekeywords={commit,Author:,Date:,chore,doc,edit,feat,fix,polish,style,tests,}, - sensitive=true, - } - \lstdefinelanguage{gitlog}{ - morekeywords={chore,doc,edit,feat,fix,polish,style,tests,}, - morecomment=[s]{[}{]}, % chktex 9 - sensitive=true, - } - - \usepackage[shortlabels]{enumitem} - \usepackage[obeyFinal,textsize=scriptsize,shadow]{todonotes} - \usepackage{textcomp} - \usepackage{multirow} - \usepackage{ellipsis} % don't use ugly unicode ellipsis - % Tiny optimizations: - \usepackage{mathtools} - \usepackage{microtype} - \usepackage{xstring} - \usepackage{wrapfig} - \ifevanbritish\usepackage[cleanlook,british]{isodate}\fi % day-first date - - % a list I like for walkthrough's --- Drew-style parts - \newlist{walk}{enumerate}{3} - \setlist[walk]{label=\bfseries (\alph*)} - % list item for MO style rubrics - \newcommand{\worth}[1]{\def\hfill{\hskip 20pt plus 1fill}\dotfill% - \IfEq{#1}{0}{\textbf{0~points}}% - {\textbf{\color{blue}#1~point\IfEndWith{#1}{1}{}{s}}}% - \par} - \newcommand{\subworth}[1]{\def\hfill{\hskip 20pt plus 1fill}\dotfill% - \IfEq{#1}{0}{{\footnotesize0~points}}% - {\textbf{\footnotesize#1~point\IfEndWith{#1}{1}{}{s}}}% - \par} - \newlist{rubric}{enumerate}{2} - \setlist[rubric,1]{label=\Roman*.} - \setlist[rubric,2]{label=(\Roman{rubrici}.\alph*)} - \fi - - %%fakesection \maketitle configuration - \@ifundefined{KOMAClassName}% - {} % do nothing outside KOMA class - {% If KOMA exists. . . - \addtokomafont{subtitle}{\Large} - \setkomafont{author}{\Large\scshape} - \setkomafont{date}{\Large\normalsize} - } - \providecommand{\thetitle}{\@title} - \providecommand{\theauthor}{\@author} - \providecommand{\thedate}{\@date} - - %%fakesection Commutative diagrams support - \ifevandiagrams - \usepackage{tikz-cd} - \usetikzlibrary{decorations.pathmorphing} - \fi - - %%fakesection Page setup - \ifevanfancy - \@ifundefined{KOMAClassName} - { - \usepackage{fancyhdr} - \setlength{\headheight}{0.75in} - \setlength{\oddsidemargin}{0in} - \setlength{\evensidemargin}{0in} - \setlength{\voffset}{-1.0in} - \setlength{\headsep}{10pt} - \setlength{\textwidth}{6.5in} - \setlength{\headwidth}{6.5in} - \setlength{\textheight}{8.75in} - \setlength{\parskip}{1ex plus 0.5ex minus 0.2ex} - \setlength{\footskip}{0.3in} - \ifevanhdr - \renewcommand{\headrulewidth}{0.5pt} - \renewcommand{\footrulewidth}{0.0pt} - \pagestyle{fancy} - \lhead{Gabriel Antonio Videtta} - \chead{} - \rhead{\nouppercase{\leftmark}} - \lfoot{} - \cfoot{\thepage} - \rfoot{} - \fi - } - { - \usepackage[headsepline]{scrlayer-scrpage} - \renewcommand{\headfont}{} - \addtolength{\textheight}{3.14cm} - \setlength{\footskip}{0.5in} - \setlength{\headsep}{10pt} - \ihead{\footnotesize\textbf{\theauthor} --- \thedate} - \automark{section} - \chead{} - \ohead{\footnotesize\textbf{\thetitle}} - \cfoot{\pagemark} - } - \fi - - %%fakesection Unicode - \ifevanpkg - \ifluatex - \usepackage{fontspec} - % https://tex.stackexchange.com/a/572220/76888 - \directlua{luaotfload.add_fallback - ("evans_fallbacks", - { - "NotoColorEmoji:mode=harf;", - "Source Han Sans TW:style=Regular;", - "Noto Serif CJK SC:style=Regular;", - } - )} - \setmainfont{lmroman10-regular}[ - BoldFont=lmroman10-bold, - ItalicFont=lmroman10-italic, - BoldItalicFont=lmroman10-bolditalic, - SlantedFont=lmromanslant10-regular, - BoldSlantedFont=lmromanslant10-bold, - SmallCapsFont=lmromancaps10-regular, - RawFeature={fallback=evans_fallbacks} - ] - \setsansfont{lmsans10-regular}[ - BoldFont=lmsans10-bold, - ItalicFont=lmsans10-oblique, - BoldItalicFont=lmsans10-boldoblique, - RawFeature={fallback=evans_fallbacks} - ] - \else - \ifevanchinese - \usepackage[italian]{babel} - \usepackage[encapsulated]{CJK} - \usepackage{ucs} - \usepackage[utf8x]{inputenc} - \newenvironment{bsmi}{\begin{CJK}{UTF8}{bsmi}}{\end{CJK}} - \newcommand{\cn}[1]{\begin{bsmi}#1\end{bsmi}} - \AtBeginDocument{\begin{CJK}{UTF8}{bsmi}} - \AtEndDocument{\end{CJK}} - \fi - \fi - \fi - - %%fakesection Hints - \ifevanhints - \usepackage{answers} - \Newassociation{hint}{hintitem}{hints} - \renewcommand{\solutionextension}{out} - \Opensolutionfile{hints} - \newcommand{\makehints}{\Closesolutionfile{hints}\input{hints.out}} % chktex 27 - \fi -%%fakesection END MAIN SETUP -\fi \ No newline at end of file +\declaretheoremstyle[ + headfont=\sffamily\bfseries\color{MidnightBlue}, + mdframed={style=mdbluebox}, + headpunct={\\[3pt]}, + postheadspace={0pt} +]{thmbluebox} + +\mdfdefinestyle{mdbluebox2}{% + roundcorner=10pt, + linewidth=1pt, + skipabove=12pt, + innerbottommargin=9pt, + skipbelow=2pt, + linecolor=blue, + nobreak=true, + backgroundcolor=BlueViolet!9, +} +\declaretheoremstyle[ + headfont=\sffamily\bfseries\color{RoyalPurple}, + mdframed={style=mdbluebox2}, + headpunct={\\[3pt]}, + postheadspace={0pt} +]{thmbluebox2} + +\mdfdefinestyle{mdredbox}{% + linewidth=0.5pt, + skipabove=12pt, + frametitleaboveskip=5pt, + frametitlebelowskip=0pt, + skipbelow=2pt, + frametitlefont=\bfseries, + innertopmargin=4pt, + innerbottommargin=8pt, + nobreak=true, + backgroundcolor=Salmon!5, + linecolor=RawSienna, +} +\declaretheoremstyle[ + headfont=\bfseries\color{RawSienna}, + mdframed={style=mdredbox}, + headpunct={\\[3pt]}, + postheadspace={0pt}, +]{thmredbox} + +\mdfdefinestyle{mdredbox2}{% + roundcorner=10pt, + linewidth=1pt, + skipabove=12pt, + innerbottommargin=9pt, + skipbelow=2pt, + linecolor=red, + nobreak=true, + backgroundcolor=WildStrawberry!5, +} +\declaretheoremstyle[ + headfont=\sffamily\bfseries\color{Maroon}, + mdframed={style=mdredbox2}, + headpunct={\\[3pt]}, + postheadspace={0pt} +]{thmredbox2} + +\mdfdefinestyle{mdgreenbox}{% + skipabove=8pt, + linewidth=2pt, + rightline=false, + leftline=true, + topline=false, + bottomline=false, + linecolor=ForestGreen, + backgroundcolor=ForestGreen!5, +} +\declaretheoremstyle[ + headfont=\bfseries\sffamily\color{ForestGreen!70!black}, + bodyfont=\normalfont, + spaceabove=2pt, + spacebelow=1pt, + mdframed={style=mdgreenbox}, + headpunct={ --- }, +]{thmgreenbox} + +\mdfdefinestyle{mdgreenbox2}{% + roundcorner=10pt, + linewidth=1pt, + skipabove=12pt, + innerbottommargin=9pt, + skipbelow=2pt, + linecolor=ForestGreen, + nobreak=true, + backgroundcolor=ForestGreen!5, +} +\declaretheoremstyle[ + headfont=\sffamily\bfseries\color{ForestGreen!70!black}, + mdframed={style=mdgreenbox2}, + headpunct={\\[3pt]}, + postheadspace={0pt} +]{thmgreenbox2} + +\mdfdefinestyle{mdblackbox}{% + skipabove=8pt, + linewidth=3pt, + rightline=false, + leftline=true, + topline=false, + bottomline=false, + linecolor=black, + backgroundcolor=RedViolet!5!gray!10, +} +\declaretheoremstyle[ + headfont=\bfseries, + bodyfont=\normalfont\small, + spaceabove=0pt, + spacebelow=0pt, + mdframed={style=mdblackbox} +]{thmblackbox} + +\mdfdefinestyle{mdblackbox2}{% + skipabove=8pt, + linewidth=3pt, + rightline=false, + leftline=true, + topline=false, + bottomline=false, + linecolor=gray, + backgroundcolor=RedViolet!5!gray!10, +} +\declaretheoremstyle[ + headfont=\bfseries, + bodyfont=\normalfont\small, + spaceabove=0pt, + spacebelow=0pt, + mdframed={style=mdblackbox2} +]{thmblackbox2} + +\newcommand{\listhack}{$\empty$\vspace{-2em}} + +\declaretheorem[% + style=thmbluebox,name=Teorema,numberwithin=section]{theorem} + +\declaretheorem[style=thmgreenbox2,name=Lemma,sibling=theorem]{lemma} +\declaretheorem[style=thmredbox2,name=Proposizione,sibling=theorem]{proposition} +\declaretheorem[style=thmbluebox2,name=Corollario,sibling=theorem]{corollary} +\declaretheorem[style=thmbluebox,name=Teorema,numbered=no]{theorem*} +\declaretheorem[style=thmgreenbox2,name=Lemma,numbered=no]{lemma*} +\declaretheorem[style=thmredbox2,name=Proposizione,numbered=no]{proposition*} +\declaretheorem[style=thmbluebox2,name=Corollario,numbered=no]{corollary*} +\declaretheorem[style=thmgreenbox,name=Claim,sibling=theorem]{claim} +\declaretheorem[style=thmgreenbox,name=Claim,numbered=no]{claim*} +\declaretheorem[style=thmredbox,name=Esempio,sibling=theorem]{example} +\declaretheorem[style=thmredbox,name=Esempio,numbered=no]{example*} +\declaretheorem[style=thmblackbox, name=Definizione,sibling=theorem]{definition} +\declaretheorem[style=thmblackbox, name=Definizione,numbered=no]{definition*} +\declaretheorem[style=thmblackbox2,name=Osservazione,sibling=theorem]{remark} +\declaretheorem[style=thmblackbox2,name=Osservazione,numbered=no]{remark*} +\declaretheorem[name=Congettura,sibling=theorem]{conjecture} +\declaretheorem[name=Congettura,numbered=no]{conjecture*} +\declaretheorem[name=Esercizio,sibling=theorem]{exercise} +\declaretheorem[name=Esercizio,numbered=no]{exercise*} +\declaretheorem[name=Asserzione,sibling=theorem]{fact} +\declaretheorem[name=Asserzione,numbered=no]{fact*} +\declaretheorem[name=Problema,sibling=theorem]{problem} +\declaretheorem[name=Problema,numbered=no]{problem*} +\declaretheorem[name=Domanda,sibling=theorem]{ques} +\declaretheorem[name=Domanda,numbered=no]{ques*} + +\Crefname{claim}{Claim}{Claim} +\Crefname{conjecture}{Congettura}{Congetture} +\Crefname{exercise}{Esercizio}{Esercizi} +\Crefname{fact}{Asserzione}{Asserzioni} +\Crefname{problem}{Problema}{Problemi} +\Crefname{ques}{Domanda}{Domande} + +\addtokomafont{partprefix}{\rmfamily} +\renewcommand*{\partformat}{\color{purple} + \scalebox{2.5}{\thepart}\enlargethispage{2em}} + +\renewcommand*{\sectionformat}% +{\color{purple}\S\thesection\enskip} +\renewcommand*{\subsectionformat}% +{\color{purple}\S\thesubsection\enskip} +\renewcommand*{\subsubsectionformat}% +{\color{purple}\S\thesubsubsection\enskip} +\KOMAoptions{numbers=noenddot} + +\usepackage{listings} +\usepackage{mathrsfs} +\usepackage{textcomp} +\lstset{basicstyle=\ttfamily\scriptsize, + backgroundcolor=\color{green!2!white}, + breakatwhitespace=true, + breaklines=true, + commentstyle=\color{green!70!black}, + frame=shadowbox, + frame=single, + identifierstyle=\color{green!20!black}, + keywordstyle=\bfseries, + keywordstyle=\bfseries\color{blue}, + numbers=left, + numbersep=5pt, + numberstyle=\tiny\sffamily\itshape\color{black!60}, + rulecolor=\color{blue!70!black}, + rulesepcolor=\color{blue!30!black}, + showstringspaces=false, + stringstyle=\color{orange}, + tabsize=4, +} + +\usepackage[shortlabels]{enumitem} +\usepackage[obeyFinal,textsize=scriptsize,shadow]{todonotes} +\usepackage{textcomp} +\usepackage{multirow} +\usepackage{ellipsis} +\usepackage{mathtools} +\usepackage{microtype} +\usepackage{xstring} +\usepackage{wrapfig} + +\addtokomafont{subtitle}{\Large} +\setkomafont{author}{\Large\scshape} +\setkomafont{date}{\Large\normalsize} + +\providecommand{\thetitle}{\@title} +\providecommand{\theauthor}{\@author} +\providecommand{\thedate}{\@date} + +\usepackage[headsepline]{scrlayer-scrpage} +\renewcommand{\headfont}{} +\addtolength{\textheight}{3.14cm} +\setlength{\footskip}{0.5in} +\setlength{\headsep}{10pt} + +\ihead{\footnotesize\textbf{\rightmark}} +\ohead{\footnotesize\textit{\thetitle}} + +\automark{section} +\chead{} +\cfoot{\pagemark} + +\hfuzz=\maxdimen +\tolerance=10000 +\hbadness=10000