From a01798d08b798ebe48ddae30f22ce628d9dcadb6 Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 12 Jan 2023 22:51:12 +0100 Subject: [PATCH] feat(algebrario): aggiunge il teorema cinese del resto --- ...0. Teoremi rilevanti sui campi finiti.tex} | 0 Aritmetica/2. Anelli euclidei, PID e UFD.tex | 153 ++++++++++++++++++ Aritmetica/7. Estensioni algebriche di K.tex | 77 ++++++++- Aritmetica/8. Campi di spezzamento.tex | 75 --------- ...ntale dell'algebra ed estensioni di Q.tex} | 2 +- ...=> 9. Introduzione a teoria dei campi.tex} | 0 Aritmetica/aritmetica.pdf | Bin 846031 -> 856322 bytes Aritmetica/aritmetica.tex | 12 +- 8 files changed, 233 insertions(+), 86 deletions(-) rename Aritmetica/{11. Teoremi rilevanti sui campi finiti.tex => 10. Teoremi rilevanti sui campi finiti.tex} (100%) delete mode 100644 Aritmetica/8. Campi di spezzamento.tex rename Aritmetica/{9. Teorema fondamentale dell'algebra ed estensioni di Q.tex => 8. Teorema fondamentale dell'algebra ed estensioni di Q.tex} (98%) rename Aritmetica/{10. Introduzione a teoria dei campi.tex => 9. Introduzione a teoria dei campi.tex} (100%) diff --git a/Aritmetica/11. Teoremi rilevanti sui campi finiti.tex b/Aritmetica/10. Teoremi rilevanti sui campi finiti.tex similarity index 100% rename from Aritmetica/11. Teoremi rilevanti sui campi finiti.tex rename to Aritmetica/10. Teoremi rilevanti sui campi finiti.tex diff --git a/Aritmetica/2. Anelli euclidei, PID e UFD.tex b/Aritmetica/2. Anelli euclidei, PID e UFD.tex index da70d80..560103e 100644 --- a/Aritmetica/2. Anelli euclidei, PID e UFD.tex +++ b/Aritmetica/2. Anelli euclidei, PID e UFD.tex @@ -490,3 +490,156 @@ euclidei. verificando la tesi. \end{proof} +\subsection{Il teorema cinese del resto} + + Il noto \nameref{th:cinese} è un risultato più generale di quanto + si sia visto nel contesto dell'aritmetica modulare. Difatti, esso è + applicabile in forma estesa a tutti gli anelli euclidei, non solo ai + numeri interi (che comunque rimangono un esempio classico di anello euclideo). \\ + + \begin{lemma} + \label{lem:pre_cinese} + + Sia $a$ un elemento riducibile di un anello euclideo $E$ e + sia $a=bc$, dove $\MCD(b, c) \in E^*$. Allora vale + il seguente isomorfismo: + + \[ A/(a) \cong A/(b) \times A/(c). \] + \end{lemma} + + \begin{proof} + Si consideri la funzione $\pi$ definita nel seguente + modo: + + \[ \pi : A/(a) \to A/(b) \times A/(c),\,e + (a) \mapsto (e + (b), e + (c)). \] + + \vskip 0.1in + + Si verifica che $\pi$ è un omomorfismo. Infatti + $\pi(1 + (a)) = (1 + (b), 1 + (c))$. \\ + + Siano $e$, $k \in A$. Allora + $\pi$ soddisfa la linearità: + + \begin{multline*} + \pi\Bigl(\bigl(e + (a)\bigr) + \bigl(k + (a)\bigr)\Bigr) = \pi\bigl(e + k + (a)\bigr) = + \bigl(e + k + (b), e + k + (c)\bigr) = + \bigl(e + (b), e + (c)\bigr) + \\ \bigl(k + (b), k + (c)\bigr) = \pi\bigl(e + (a)\bigr) + + \pi\bigl(k + (a)\bigr). + \end{multline*} + + \vskip 0.1in + + e la moltiplicatività: + + \begin{multline*} + \pi\Bigl(\bigl(e + (a)\bigr) \cdot \bigl(k + (a)\bigr)\Bigr) = \pi\bigl(ek + (a)\bigr) = + \bigl(ek + (b), ek + (c)\bigr) = + \bigl(e + (b), e + (c)\bigr) \cdot \\ \bigl(k + (b), k + (c)\bigr) = \pi\bigl(e + (a)\bigr) \cdot + \pi\bigl(k + (a)\bigr). + \end{multline*} + + \vskip 0.1in + + Si studia $\Ker \pi$ per dimostrare l'iniettività di $\pi$. + Si pone dunque $\pi\bigl(e + (a)\bigr) = \bigl(0 + (b), 0 + (c)\bigr)$. + Questa condizione è equivalente ad asserire che sia $b$ che $c$ dividano + $e$. \\ + + Sia allora $k \in E$ tale che $e=bk$. Dal momento che $c$ divide $e$, si + $e$ divide $bk$. Allora, dacché per ipotesi $\MCD(a, b) \in E^*$, per la + \propref{prop:divisione_gcd} $c$ divide $k$. Quindi esiste $j \in E$ tale che + $k=cj$. In particolare, unendo le due condizioni si ottiene + $e=bk=bcj=aj$. Pertanto $a$ divide $e$, da cui si deduce che $e + (a)$ + è equivalente a $0 + (a)$. Allora, poiché $\Ker \pi = (0)$, $\pi$ è un + monomorfismo. \\ + + Si studia invece adesso la surgettività di $\pi$. Siano $\alpha$, + $\beta \in E$. Si pone dunque $\pi\bigl(e + (a)\bigr) = + \bigl(\alpha + (b), \beta + (c)\bigr)$. Questa condizione è equivalente + al seguente sistema: + + \[ \begin{cases} e = \alpha + bk, \\ e = \beta + cj, \end{cases} \quad \text{con } k, j \in E. \] + + \vskip 0.1in + + Unendo le due condizioni si ottiene la seguente equazione: + + \[ \alpha + bk = \beta + cj \iff cj - bk = \alpha - \beta. \] + + \vskip 0.1in + + Si consideri ora $d = \MCD(b, c)$. Per l'\nameref{th:bezout} esistono + $x$, $y$ tali che: + + \[ cx+by=d, \] + + \vskip 0.1in + + da cui si ricava che: + + \[ (\alpha-\beta)(cx + by) = (\alpha-\beta)d \implies cxd\inv(\alpha-\beta)+byd\inv(\alpha-\beta)=\alpha-\beta, \] + + \vskip 0.1in + + ponendo allora $j=xd\inv(\alpha-\beta)$ e $k=-yd\inv(\alpha-\beta)$ + si ricava una possibile soluzione per $e$. Quindi + $\pi$ è un epimorfismo. \\ + + Poiché $\pi$ è sia un monomorfismo che un epimorfismo, si conclude + che $\pi$ è un isomorfismo, da cui la tesi. + + \end{proof} + + \begin{theorem}[\textit{Teorema cinese del resto}] + \label{th:cinese} + + Sia $a$ un elemento di un anello euclideo $A$ e sia + $p_1^{m_1}p_2^{m_2}\cdots p_n^{m_n}$ una sua fattorizzazione + in irriducibili non associati. + Allora vale il seguente isomorfismo: + + \[ A/(a) \cong A/(p_1^{m_1}) \times \cdots \times A/(p_n^{m_n}). \] + \end{theorem} + + \begin{proof} + Si dimostra il teorema applicando il principio di induzione su $n$, + il numero di fattori irriducibili distinti che appaiono + nella fattorizzazione di $a$. \\ + + \,(\textit{passo base}) \, Se $a$ consta di un solo fattore irriducibile, + allora banalmente $A/(a) \cong A/(p_1^{m_1})$. \\ + + \,(\textit{passo induttivo}) \, Possiamo riscrivere $a$ come + il prodotto di $(p_1^{m_1}\cdots p_{n-1}^{m_{n-1}})$ e di + $p_n^{m_n}$. \\ + + Si nota innanzitutto che $d = \MCD(p_1^{m_1}\cdots p_{n-1}^{m_{n-1}}, p_n^{m_n})$ + è un invertibile. Se così non fosse, infatti, si potrebbe + considerare un irriducibile $q$ della fattorizzazione di $d$: + tale $q$, in quanto primo per il \thref{th:irriducibili_primi}, + deve dividere un $p_j$ con $1 \leq j \leq n-1$, così + come deve dividere $p_n$. Allora $p_j$ e $q$ sono associati, + così come $q$ e $p_n$. Dunque anche $p_j$ e $p_n$ sono associati. + Tuttavia questo è un assurdo, dal momento che per ipotesi + la fattorizzazione di $a$ include irriducibili distinti e + non associati, \Lightning{}. \\ + + Allora dal \lemref{lem:pre_cinese} si ricava che: + + \[ A/(a) \cong A/(p_1^{m_1}\cdots p_{n-1}^{m_{n-1}}) \times A/(p_n^{m_n}), \] + + \vskip 0.1in + + mentre dal passo induttivo si sa già che: + + \[ A/(p_1^{m_1}\cdots p_{n-1}^{m_{n-1}}) \cong A/(p_1^{m_1}) \times \cdots \times A/(p_{n-1}^{m_{n-1}}). \] + + \vskip 0.1in + + Pertanto, unendo le due informazioni, si verifica la tesi: + + \[ A/(a) \cong + A/(p_1^{m_1}) \times \cdots \times A/(p_{n-1}^{m_{n-1}}) \times A/(p_n^{m_n}). \] + + \end{proof} diff --git a/Aritmetica/7. Estensioni algebriche di K.tex b/Aritmetica/7. Estensioni algebriche di K.tex index 4969aad..1855c08 100644 --- a/Aritmetica/7. Estensioni algebriche di K.tex +++ b/Aritmetica/7. Estensioni algebriche di K.tex @@ -519,4 +519,79 @@ seguente teorema. $A(c)$ è un'estensione finita di $A$, ed in quanto tale, per la \propref{prop:estensione_finita_algebrica}, è anche algebrica. Quindi $c$ è algebrico su $A$, da cui la tesi. -\end{proof} \ No newline at end of file +\end{proof} + +\begin{theorem} + \label{th:esistenza_spezzamento} + Sia $A$ un campo, e sia $f(x) \in A[x]$. + Allora esiste sempre un estensione di $A$ in cui siano + contenute tutte le radici di $f(x)$. +\end{theorem} + +\begin{proof} + Si dimostra il teorema applicando il principio di induzione sul + grado di $f(X)$. \\ + + \ (\textit{passo base}) \,Sia $\deg f(x) = 0$. Allora $A$ stesso è un + campo in cui sono contenute tutte le radici, dacché esse non esistono. \\ + + \ (\textit{passo induttivo}) \,Sia $\deg f(x) = n$. Sia $f_1(x)$ un + irriducibile di $f(x)$ e sia $\gamma(x) \in A[x]$ tale che + $f(x)=f_1(x)\gamma(x)$. Allora, per il \thref{th:campo_quoziente_irriducibile} + $A[x]/(f_1(x))$ è un campo, in cui, per la \propref{prop:radice_quoziente}, + $f_1(x)$ ammette radice. \\ + + Poiché $\deg \gamma(x) < n$, per il passo induttivo + esiste un campo $C$ che estende $A[x]/(f_1(x))$ in cui risiedono tutte le sue radici. Dacché $C$ contiene $A[x]/(f_1(x))$, sia le radici + di $f_1(x)$ che di $\gamma(x)$ risiedono in $C$. Tuttavia queste sono + tutte le radici di $f(x)$, si conclude che $C$, che è un'estensione di $A[x]/(f_1(x))$, e quindi anche di $A$, è il campo ricercato. +\end{proof} + +\subsection{Campi di spezzamento di un polinomio} + +Pertanto ora è possibile enunciare la definizione di \textit{campo di spezzamento}. + +\begin{definition} + Si definisce \textbf{campo di spezzamento} di un polinomio $f(x) \in A[x]$ un + campo $C$ con le seguenti caratteristiche: + + \begin{itemize} + \item $f(x)$ si fattorizza in $C[x]$ come prodotto di irriducibili di + primo grado (i.e. in $C[x]$ risiedono tutte le radici di $f(x)$), + \item Se $B$ è un campo tale che $A \subseteq B \subsetneq C$, allora + $f(x)$ non si fattorizza in $B[x]$ come prodotto di irriducibili di + primo grado. + \end{itemize} +\end{definition} + +\begin{remark*} + Per il \thref{th:esistenza_spezzamento} esiste sempre un campo di spezzamento + di un polinomio, dunque la definizione data è una buona definizione. +\end{remark*} + +\begin{remark*} + 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*} + +\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/8. Campi di spezzamento.tex b/Aritmetica/8. Campi di spezzamento.tex deleted file mode 100644 index 5d26b17..0000000 --- a/Aritmetica/8. Campi di spezzamento.tex +++ /dev/null @@ -1,75 +0,0 @@ -\section{Campi di spezzamento} - -\begin{theorem} - \label{th:esistenza_spezzamento} - Sia $A$ un campo, e sia $f(x) \in A[x]$. - Allora esiste sempre un estensione di $A$ in cui siano - contenute tutte le radici di $f(x)$. -\end{theorem} - -\begin{proof} - Si dimostra il teorema applicando il principio di induzione sul - grado di $f(X)$. \\ - - \ (\textit{passo base}) \,Sia $\deg f(x) = 0$. Allora $A$ stesso è un - campo in cui sono contenute tutte le radici, dacché esse non esistono. \\ - - \ (\textit{passo induttivo}) \,Sia $\deg f(x) = n$. Sia $f_1(x)$ un - irriducibile di $f(x)$ e sia $\gamma(x) \in A[x]$ tale che - $f(x)=f_1(x)\gamma(x)$. Allora, per il \thref{th:campo_quoziente_irriducibile} - $A[x]/(f_1(x))$ è un campo, in cui, per la \propref{prop:radice_quoziente}, - $f_1(x)$ ammette radice. \\ - - Poiché $\deg \gamma(x) < n$, per il passo induttivo - esiste un campo $C$ che estende $A[x]/(f_1(x))$ in cui risiedono tutte le sue radici. Dacché $C$ contiene $A[x]/(f_1(x))$, sia le radici - di $f_1(x)$ che di $\gamma(x)$ risiedono in $C$. Tuttavia queste sono - tutte le radici di $f(x)$, si conclude che $C$, che è un'estensione di $A[x]/(f_1(x))$, e quindi anche di $A$, è il campo ricercato. -\end{proof} - -Pertanto ora è possibile enunciare la definizione di \textit{campo di spezzamento}. - -\begin{definition} - Si definisce \textbf{campo di spezzamento} di un polinomio $f(x) \in A[x]$ un - campo $C$ con le seguenti caratteristiche: - - \begin{itemize} - \item $f(x)$ si fattorizza in $C[x]$ come prodotto di irriducibili di - primo grado (i.e. in $C[x]$ risiedono tutte le radici di $f(x)$), - \item Se $B$ è un campo tale che $A \subseteq B \subsetneq C$, allora - $f(x)$ non si fattorizza in $B[x]$ come prodotto di irriducibili di - primo grado. - \end{itemize} -\end{definition} - -\begin{remark*} - Per il \thref{th:esistenza_spezzamento} esiste sempre un campo di spezzamento - di un polinomio, dunque la definizione data è una buona definizione. -\end{remark*} - -\begin{remark*} - 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*} - -\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/8. Teorema fondamentale dell'algebra ed estensioni di Q.tex similarity index 98% rename from Aritmetica/9. Teorema fondamentale dell'algebra ed estensioni di Q.tex rename to Aritmetica/8. Teorema fondamentale dell'algebra ed estensioni di Q.tex index 25f5808..b6919b0 100644 --- a/Aritmetica/9. Teorema fondamentale dell'algebra ed estensioni di Q.tex +++ b/Aritmetica/8. Teorema fondamentale dell'algebra ed estensioni di Q.tex @@ -1,4 +1,4 @@ -\section{Teorema fondamentale dell'Algebra e radici reali in \texorpdfstring{$\QQx$}{Q[]}} +\section{Teorema fondamentale dell'Algebra e radici reali in \texorpdfstring{$\QQx$}{Q[x]}} Si enuncia adesso il \nameref{th:algebra}, senza tuttavia fornirne una dimostrazione\footnote{Per la dimostrazione si rimanda diff --git a/Aritmetica/10. Introduzione a teoria dei campi.tex b/Aritmetica/9. Introduzione a teoria dei campi.tex similarity index 100% rename from Aritmetica/10. Introduzione a teoria dei campi.tex rename to Aritmetica/9. 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zaoq3@sXYtK)WYx%WnyQ(9f4^>m-`R#$=Xv6!0dTV+)c12-7zhdIgy5CFALGxgdi}} z%7NLdQNXAow~3dew^iLhNKZlJTn7PY$*{oeg%@Cc&rx4A5=lGDzW_#%&hr#||jM=!C8u6bj7>B^P+AAI1_{{g(wB0>NF diff --git a/Aritmetica/aritmetica.tex b/Aritmetica/aritmetica.tex index 680a295..183d891 100644 --- a/Aritmetica/aritmetica.tex +++ b/Aritmetica/aritmetica.tex @@ -90,25 +90,19 @@ \thispagestyle{empty} ~\newpage -\include{8. Campi di spezzamento} +\include{8. Teorema fondamentale dell'algebra ed estensioni di Q} \newpage \thispagestyle{empty} ~\newpage -\include{9. Teorema fondamentale dell'algebra ed estensioni di Q} +\include{9. Introduzione a teoria dei campi} \newpage \thispagestyle{empty} ~\newpage -\include{10. Introduzione a teoria dei campi} - -\newpage -\thispagestyle{empty} -~\newpage - -\include{11. Teoremi rilevanti sui campi finiti} +\include{10. Teoremi rilevanti sui campi finiti} \newpage \thispagestyle{empty}