From 600b46565fbcd459d75dd9a6842d3747eca70388 Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 30 Nov 2023 12:10:07 +0100 Subject: [PATCH] feat(algebra1): introduce le chiusure algebriche e i campi perfetti --- .../main.pdf | Bin 0 -> 165002 bytes .../main.tex | 201 ++++++++++++++++++ 2 files changed, 201 insertions(+) create mode 100644 Secondo anno/Algebra 1/3. Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.pdf create mode 100644 Secondo anno/Algebra 1/3. Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex 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 dei campi/2. 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Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex b/Secondo anno/Algebra 1/3. Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex new file mode 100644 index 0000000..0e8ef6d --- /dev/null +++ b/Secondo anno/Algebra 1/3. Teoria dei campi/2. Chiusura algebrica di un campo e campi di spezzamento/main.tex @@ -0,0 +1,201 @@ +\documentclass[12pt]{scrartcl} +\usepackage{notes_2023} + +\begin{document} + \title{Chiusura algebrica di un campo e campi di spezzamento} + \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. + \end{note} \bigskip + + + Questo documento si propone di illustrare le principali + proprietà e caratteristiche dei campi algebricamente + chiusi, delle chiusure algebriche e dei campi di + spezzamento, col proposito di dare i mezzi necessari + per approcciarsi alla teoria di Galois. Per questo + motivo si presentano le seguenti definizioni: + + \begin{definition}[campo algebricamente chiuso] + Un campo $K$ si dice \textbf{algebricamente chiuso} + se ogni polinomio a coefficienti in $K$ ammette una + radice in $K$. Equivalentemente, $K$ è algebricamente + chiuso se ogni polinomio $p \in K[x]$ ha tutte le proprie + radici in $K$, e quindi se gli irriducibili di $K$ sono + tutti e soli i polinomi di grado unitario. + \end{definition} + + \begin{definition}[chiusura algebrica] + Un estensione $\faktor{\Omega}{K}$ si dice + \textbf{chiusura algebrica} di $K$, e si + indica usualmente con $\overline{K}$, se $\Omega$ + è un campo algebricamente chiuso e se + $\Omega$ è un'estensione algebrica su $K$. + \end{definition} + + \begin{remark} + Per esempio, una chiusura algebrica di $\RR$ è $\CC$, + per il Teorema fondamentale dell'algebra. + \end{remark} + + \begin{proposition} + Sia $\Omega$ un campo algebricamente chiuso. Se allora + $K$ è un sottocampo di $\Omega$, vale che + $K'$, il campo degli elementi algebrici + su $K$, è una chiusura algebrica di $K$. + \end{proposition} + + \begin{proof} + Chiaramente + $K'$ è un'estensione algebrica su + $K$. Si verifica allora che $K'$ è + algebricamente chiuso. Sia $p \in K'[x]$. + Dal momento che $K$ è algebricamente chiuso, e + che $p$ appartiene anche a $K[x]$, allora + $p$ ammette una radice $\alpha \in \Omega$. Si mostra che + $\alpha$ è algebrico su $K$. Poiché allora + $\faktor{K'(\alpha)}{K'}$ è + un'estensione algebrica (infatti $p$ annulla $\alpha$ + per ipotesi) e $\faktor{K'}{K}$ è algebrica + per ipotesi, allora $K'(\alpha)$ è algebrica + su $K$, e dunque $\alpha$ è algebrico su $K$, pertanto + $\alpha \in K'$, da cui la tesi. + \end{proof} + + \begin{remark} + Poiché $\QQ$ è un sottocampo di $\CC$ e $\CC$ è + un campo algebricamente chiuso, il campo degli + elementi algebrici di $\QQ$ è una chiusura algebrica di + $\QQ$ per la proposizione precedente. + \end{remark} + + Adesso si enuncia, senza dimostrarlo, un teorema su cui si baserà buona parte della prossima teoria: + + \begin{theorem}[esistenza ed unicità della chiusura algebrica] + Esiste ed è unica, a meno di $K$-isomorfismo\footnote{ + Un $K$-isomorfismo è un isomorfismo tra estensioni + di $K$ che fissa $K$, ossia che ristretto a $K$ è + l'identità di $K$. + }, + la chiusura algebrica di $K$. + \end{theorem} + + \begin{remark} + Poiché il campo degli elementi algebrici di $\QQ$ è una chiusura algebrica di + $\QQ$ ed è un insieme numerabile, $\CC$ non può + essere una chiusura algebrica di $\QQ$ dacché + $\CC$ ha la cardinalità del continuo (e dunque non possono + esistere bigezioni tra $\CC$ e $\overline{\QQ}$). Poiché + $\CC$ è però algebricamente chiuso, può solamente + verificarsi che $\CC$ non sia un'estensione algebrica + di $\QQ$. Più facilmente, $\pi \in \RR$ non è algebrico su $\QQ$, + e così né $\RR$ né $\CC$ sono estensioni algebriche su $\QQ$. + \end{remark} + + \begin{definition}[campo di spezzamento] + Sia $\mathcal{F}$ una famiglia di polinomi di $K[x]$. + Si definisce allora \textbf{campo di spezzamento} di + $\mathcal{F}$ una estensione $F$ di $K$ tale per cui: + + \begin{itemize} + \item ogni $p \in \mathcal{F}$ si decompone in fattori lineari in + $F[x]$, + \item se $L$ è un'estensione su $K$ tale per cui + $L \subsetneq F$, allora esiste $p \in \mathcal{F}$ + non si decompone in fattori lineari in $L[x]$. + \end{itemize} + + Equivalentemente $F$ è un'estensione minimale in cui + ogni polinomio di $\mathcal{F}$ si decompone in fattori + lineari. + \end{definition} + + Come per le chiusure algebriche, si enuncia il seguente + teorema senza dimostrazione\footnote{ + L'esistenza di un campo di spezzamento è piuttosto + facile da dimostrare, è sufficiente considerare + l'estensione di $K$ a cui si aggiungono tutte le + radici del polinomio. + }: + + \begin{theorem}[esistenza ed unicità del campo di spezzamento] + Esiste ed è unico, a meno di $K$-isomorfismo, il + campo di spezzamento di $\mathcal{F}$ su $K$. + \end{theorem} + + \begin{definition}[coniugati di $\alpha$] + Se $\alpha \in \faktor{L}{K}$ è algebrico su $K$, si definiscono \textbf{coniugati} di $\alpha$ su $K$ le + radici di $\mu_\alpha$ su $K$. + \end{definition} + + I coniugati di $\alpha$ sono speciali in quanto + permettono di studiare + le $K$-immersioni di $K(\alpha)$ in $\overline{K}$, ossia + di studiare i campi $K$-isomorfi a $K(\alpha)$ presenti in + $\overline{K}$, come dimostra la: + + \begin{proposition} + Sia $\alpha \in \faktor{L}{K}$ algebrico su $K$. Allora, + se $d$ è il numero di coniugati distinti di $\alpha$, + esistono esattamente $d$ $K$-immersioni di $K(\alpha)$ + in $\overline{K}$ e sono tali da mandare $\alpha$ in + un suo altro coniugato. + \end{proposition} + + \begin{proof} + Per considerare le $K$-immersioni di $K(\alpha)$ in + $K$, si considera prima l'isomorfismo: + \[ K(\alpha) \cong K[x] \quot{(\mu_\alpha)}. \] + Per il Primo teorema di isomorfismo, esistono + allora tanti omomorfismi da $K(\alpha)$ in $\overline{K}$ + quanti sono gli omomorfismi da $K[x]$ in $\overline{K}$ che + annullano $(\mu_\alpha)$. Un omomorfismo $\varphi$ + da $K[x]$ a $\overline{K}$ che fissa $K$ è completamente determinato da + $\beta = \varphi(x)$ ed in particolare mappa $p \in K[x]$ + a $p(\beta)$. Affinché allora $(\mu_\alpha)$ + appartenga a $\Ker \varphi$, $\mu_\alpha(\beta) = 0$, e quindi + $\beta$ deve essere un coniugato di $\alpha$. Pertanto + gli omomorfismi da $K(\alpha)$ a $\overline{K}$ sono + tali per cui $\alpha$ venga mandato in $\beta$. Questi + sono $K$-immersioni dal momento che l'unità viene preservata, + da cui la tesi. + \end{proof} + + \hr + + \begin{definition}[estensione separabile] + Un'estensione $\faktor{L}{K}$ si dice \textbf{separabile} + se per ogni $\alpha \in L$, $\mu_{\alpha,K}$ ha radici + distinte. + \end{definition} + + \begin{definition}[campo perfetto] + Un campo si dice \textbf{perfetto} se le derivate dei + suoi polinomi irriducibili non sono mai nulle. + Equivalentemente un campo è perfetto se i suoi + polinomi irriducibili hanno sempre radici distinte. + \end{definition} + + \begin{remark} + Le estensioni di un campo perfetto sono sempre separabili. + Infatti il polinomio minimo su $K$ è in particolare + un irriducibile, e quindi ha radici distinti. + \end{remark} + + \begin{note} + Si assumerà d'ora in poi che \underline{\textit{$K$ + è perfetto}}, in modo tale da semplificare l'introduzione + alla teoria di Galois. + \end{note} + + \begin{remark} + Poiché $K$ è perfetto, le $K$-immersioni di $K(\alpha)$ + sono esattamente $[K(\alpha) : K] = \deg_K \alpha$. + \end{remark} +\end{document} \ No newline at end of file