From 58fc8303d586adf23aaa03eb0c57b74e009fb44f Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 6 Dec 2023 19:36:47 +0100 Subject: [PATCH] feat(algebra1): teorema dell'elemento primitivo --- .../main.pdf | Bin 221977 -> 222059 bytes .../main.tex | 2 + .../main.pdf | Bin .../main.tex | 0 .../main.pdf | Bin .../main.tex | 0 .../main.pdf | Bin 0 -> 168679 bytes .../main.tex | 98 ++++++++++++++++++ 8 files changed, 100 insertions(+) rename Secondo anno/Algebra 1/{3. Teoria dei campi => 3. Teoria delle estensioni di campo e di Galois}/1. Estensioni di campo ed elementi algebrici e trascendenti/main.pdf (97%) rename Secondo anno/Algebra 1/{3. Teoria dei campi => 3. Teoria delle estensioni di campo e di Galois}/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex (99%) rename Secondo anno/Algebra 1/{3. Teoria dei campi => 3. Teoria delle estensioni di campo e di Galois}/2. Chiusura algebrica di un campo e campi di spezzamento/main.pdf (100%) rename Secondo anno/Algebra 1/{3. Teoria dei campi => 3. Teoria delle estensioni di campo e di Galois}/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex (100%) rename Secondo anno/Algebra 1/{3. Teoria dei campi => 3. Teoria delle estensioni di campo e di Galois}/3. Estensioni normali e gruppo di Galois/main.pdf (100%) rename Secondo anno/Algebra 1/{3. Teoria dei campi => 3. Teoria delle estensioni di campo e di Galois}/3. Estensioni normali e gruppo di Galois/main.tex (100%) create mode 100644 Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.pdf create mode 100644 Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.tex diff --git a/Secondo anno/Algebra 1/3. Teoria dei campi/1. Estensioni di campo ed elementi algebrici e trascendenti/main.pdf b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/1. 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Teoria dei campi/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex similarity index 99% rename from Secondo anno/Algebra 1/3. Teoria dei campi/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex rename to Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex index 4b2326e..b00031a 100644 --- a/Secondo anno/Algebra 1/3. Teoria dei campi/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex +++ b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex @@ -66,6 +66,8 @@ \begin{definition}[estensione semplice] Un'estensione $\faktor{L}{K}$ si dice \textbf{semplice} se esiste $\alpha \in L$ tale per cui $L = K(\alpha)$. + Tale $\alpha$ si definisce \textbf{elemento primitivo} + di $L$ su $K$. \end{definition} \begin{remark} diff --git a/Secondo anno/Algebra 1/3. Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.pdf b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/2. Chiusura algebrica di un campo e campi di spezzamento/main.pdf similarity index 100% rename from Secondo anno/Algebra 1/3. Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.pdf rename to Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/2. Chiusura algebrica di un campo e campi di spezzamento/main.pdf diff --git a/Secondo anno/Algebra 1/3. Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex similarity index 100% rename from Secondo anno/Algebra 1/3. Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex rename to Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex diff --git a/Secondo anno/Algebra 1/3. Teoria dei campi/3. Estensioni normali e gruppo di Galois/main.pdf b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/3. Estensioni normali e gruppo di Galois/main.pdf similarity index 100% rename from Secondo anno/Algebra 1/3. Teoria dei campi/3. Estensioni normali e gruppo di Galois/main.pdf rename to Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/3. Estensioni normali e gruppo di Galois/main.pdf diff --git a/Secondo anno/Algebra 1/3. Teoria dei campi/3. Estensioni normali e gruppo di Galois/main.tex b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/3. Estensioni normali e gruppo di Galois/main.tex similarity index 100% rename from Secondo anno/Algebra 1/3. Teoria dei campi/3. Estensioni normali e gruppo di Galois/main.tex rename to Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/3. Estensioni normali e gruppo di Galois/main.tex diff --git a/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.pdf b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. 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Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.tex b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.tex new file mode 100644 index 0000000..b60dfb6 --- /dev/null +++ b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/4. Il teorema dell'elemento primitivo e di corrispondenza di Galois/main.tex @@ -0,0 +1,98 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Il teorema dell'elemento primitivo e di corrispondenza di Galois} + \maketitle + + \begin{note} + Per $K$, $L$ ed $F$ si intenderanno sempre dei campi. + Se non espressamente detto, si sottintenderà anche + che $K \subseteq L$, $F$, e che $L$ ed $F$ sono + estensioni costruite su $K$. Per $[L : K]$ si + intenderà $\dim_K L$, ossia la dimensione di $L$ + come $K$-spazio vettoriale. Per scopi didattici, si + considerano solamente campi perfetti, e dunque estensioni che sono sempre separabili, purché + non esplicitamente detto diversamente. + \end{note} \bigskip + + Si dimostrano in questo documento i due teoremi più + importanti della teoria elementare delle estensioni di campo + e di Galois, il \textit{teorema dell'elemento primitivo} ed + il \textit{teorema di corrispondenza di Galois}. + + \begin{theorem}[dell'elemento primitivo] + Sia $\faktor{L}{K}$ un'estensione separabile e + finita. Allora $\faktor{L}{K}$ è semplice. + \end{theorem} + + \begin{proof} + Si distinguono i casi in cui $K$ è un campo finito + o infinito. + + \begin{itemize} + \item[($K$ finito)\;] Poiché $K$ è finito e + $L$ è un'estensione finita su $K$, a sua volta + $L$ è un campo finito. Pertanto $L^*$ è un + sottogruppo moltiplicativo finito di un campo, ed + è pertant ciclico. Se $\alpha \in L^*$ è allora + un generatore di $L^*$, vale che $L$ è uguale a + $K(\alpha)$. Pertanto $\faktor{L}{K}$ è un'estensione + semplice. + + \item[($K$ infinito)\;] + Si fornisce una dimostrazione costruttiva del + teorema, che permette di trovare algoritmicamente + un elemento primitivo per $L$. + Poiché $L$ è un'estensione + finita di $K$, $L$ è finitamente generato da + elementi algebrici su $K$. \medskip + + + Sia allora + $L = K(\alpha_1, \ldots, \alpha_n)$, dove + $\{ \alpha_i \}$ è una base di + $\faktor{L}{K}$ come $K$-spazio. È sufficiente + che $K(\alpha_1, \alpha_2)$ sia semplice affinché + anche $L$ lo sia. Infatti + si dimostrerebbe che $K(\alpha_1, \alpha_2) = + K(\gamma)$ per qualche $\gamma \in K(\alpha_1, \alpha_2)$, + e quindi $K(\alpha_1, \ldots, \alpha_n) = + K(\gamma, \alpha_3, \ldots, \alpha_n)$. Reiterando + allora il processo su $K(\gamma, \alpha_3)$ si + troverà un elemento primitivo, e così, induttivamente, + si dimostra che in particolare $L$ è semplice. Se + invece $n = 1$, la tesi è ovvia. \medskip + + + Sia allora, senza perdita di generalità, $L = K(\alpha, \beta)$. Sia $[L : K] = n$. Allora, poiché $L$ + è un'estensione separabile su $K$, esistono + esattamente $n$ distinte $K$-immersioni di $L$, + dette $\varphi_i$. + Si definisca allora $p(x) \in \overline{K}[x]$ tale per cui: + \[ p(x) = \prod_{1 \leq i < j \leq n} (x \varphi_i(\alpha) + \varphi_i(\beta) - x \varphi_j(\alpha) - \varphi_j(\beta)). \] + Si dimostra che $p(x)$ non è nullo. Infatti, + se lo fosse, almeno uno dei fattori della produttoria + dovrebbe essere nullo. In tal caso si avrebbe + $\varphi_i(\alpha) = \varphi_j(\alpha)$ e + $\varphi_i(\beta) = \varphi_j(\beta)$, e dunque + $\varphi_i \equiv \varphi_j$, benché $i \neq j$, + \Lightning. Allora $\deg p = \binom{n}{2} > 0$. + Dal momento che $K$ è infinito, esiste\footnote{ + A livello algoritmico è sufficiente valutare + $p(x)$ in al più $n+1$ valori distinti in $K$ + per ottenere un $x$ funzionale per la tesi. + } $t \in K$ + tale per cui $p(t) \neq 0$. \medskip + + + Detto $\gamma = \alpha t + \beta$, $\gamma$ + ha esattamente $n$ coniugati. Infatti + $\varphi_i(\gamma) \neq \varphi_j(\gamma)$ $\forall i < j$, altrimenti $\gamma$ annullerebbe $p(x)$. Pertanto + $[K(\gamma) : K] = n = [K(\alpha, \beta) : K]$, + da cui $K(\alpha, \beta) = K(\gamma)$, ossia la tesi. + \end{itemize} + \end{proof} + + %TODO: aggiungere corrispondenza di Galois +\end{document} \ No newline at end of file