From 8c8b130d4921bac53938ef6531e7a338986bbdba Mon Sep 17 00:00:00 2001 From: Hearot Date: Thu, 30 Nov 2023 10:15:21 +0100 Subject: [PATCH] feat(algebra1): termina la parte relativa alle estensioni di campo --- .../main.pdf | Bin 186991 -> 221977 bytes .../main.tex | 185 +++++++++++++++--- 2 files changed, 158 insertions(+), 27 deletions(-) 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 dei campi/1. 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EFDFs1l>h($ diff --git 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 dei campi/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex index e687df6..4b2326e 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 dei campi/1. Estensioni di campo ed elementi algebrici e trascendenti/main.tex @@ -60,7 +60,7 @@ Sia $\alpha \in K$. Allora si definisce $K(\alpha)$ come la più piccola estensione di $K$ che contiene $\alpha$, ossia: - \[ K(\alpha) = \bigcap_{\substack{\faktor{F_i}{K} \text{ campo} \\ \alpha \in F_i}} F_i. \] + \[ K(\alpha) = \bigcap_{\substack{\faktor{F_i}{K} \text{ campo} \\[0.02in] \alpha \in F_i}} F_i. \] \end{definition} \begin{definition}[estensione semplice] @@ -197,6 +197,68 @@ \[ LM = K(\alpha_1, \ldots, \alpha_m, \beta_1, \ldots, \beta_n). \] \end{remark} + + \begin{theorem}[delle torri algebriche] + Siano $K \subseteq L \subseteq F$ campi. Allora $\faktor{F}{K}$ è un'estensione finita se e solo se + $\faktor{L}{K}$ e $\faktor{F}{L}$ sono estensioni finite. \medskip + + + In particolare\footnote{Si può generalizzare questa formula ad $F$ spazio vettoriale su $K$ e $L$ con $K \subseteq L$, a patto che $K$ e $L$ siano campi.}, se $\faktor{F}{K}$ è un'estensione finita, + vale che: + \[ [F : K] = [F : L] [L : K]. \] + \end{theorem} + + \begin{proof} + Se $\faktor{F}{K}$ è un'estensione finita, allora a + maggior ragione $\faktor{L}{K}$ è un'estensione finita + dal momento che $L$ è un $K$-sottospazio vettoriale di + $F$, che è un $K$-spazio vettoriale. Inoltre, + anche $\faktor{F}{L}$ è un'estensione finita, dacché + una base di $\faktor{F}{K}$ è un insieme di generatori + su $\faktor{F}{L}$, dal momento che $K \subseteq L$. \medskip + + + Si mostra adesso che se $\faktor{L}{K}$ e $\faktor{F}{L}$ sono estensioni finite, allora $F$ è uno spazio + finito-dimensionale su $K$ e vale che: + \[ [F : K] = [F : L] [L : K]. \] + Siano $[F : L] = m$ e $[L : K] = n$. Sia + $\BB_F = (f_1, \ldots, f_m)$ una base + di $F$ su $L$, e sia $\BB_L = (l_1, \ldots, l_n)$ una + base di $L$ su $K$. \\ + + Si dimostra che la seguente è una base di $F$ su $K$: + \[\BB_F \BB_L = \{ f_1 l_1, \ldots, f_1 l_n, \ldots, f_m l_n\}, \] + dove si osserva che $\abs{\BB_F \BB_L} = [F : L] [L : K]$. + Si mostra innanzitutto che $\BB_F \BB_L$ è un insieme + di generatori. Sia $f \in F$. + Allora si può scrivere $f$ come combinazione lineare + finita con scalari in $L$: + \[f = \sum_{i=1}^m \beta_i f_i.\] + A sua volta, allora, si può scrivere ogni $\beta_i \in L$ + come combinazione lineare finita con scalari in $K$: + \[\beta_i = \sum_{j=1}^n \gamma_j^{(i)} l_j.\] + Combinando queste due identità, si verifica che + $\BB_F \BB_L$ genera $F$ come $K$-spazio vettoriale: + \[ f = \sum_{i=1}^m \sum_{j=1}^n \gamma_j^{(i)} \, l_j f_i. \] + + + Infine, si verifica che $\BB_F \BB_L$ è un insieme linearmente + indipendente. Si consideri l'equazione: + + \[ \sum_{i=1}^m \sum_{j=1}^n \gamma_j^{(i)} \, l_j f_i = \sum_{i=1}^m \left(\sum_{j=1}^n \gamma_j^{(i)} l_j \right) f_i = 0. \] + + Poiché $\BB_F$ è linearmente indipendente, si deduce + che: + + \[ \sum_{j=1}^n \gamma_j^{(i)} l_j = 0. \] + + Tuttavia, $\BB_L$ è a sua volta linearmente indipendente, + e quindi $\gamma_j^{(i)} = 0$, $\forall i, j$. Dunque + $\BB_F \BB_L$ è linearmente indipendente, e quindi è + una base dacché è anche un insieme di generatori per + $F$ come $K$-spazio vettoriale. Pertanto $F$ è un'estensione + finita di $K$ e vale la tesi. + \end{proof} \begin{proposition} Siano $L$ e $M$ due campi tali per cui @@ -256,26 +318,28 @@ (se $\alpha \neq 0$) e $\beta\inv$ (se $\beta \neq 0$) sono elementi algebrici di $K$, e quindi $A$ è un campo, e a maggior ragione un'estensione algebrica di $K$. \end{proof} - + + Le estensioni finite sono completamente caratterizzate + in qualità di estensioni finitamente generate da elementi + algebrici sul campo di riferimento, come mostra la: + \begin{proposition} - Se $K \subseteq L \subseteq F$ è una torre di - estensioni e $\faktor{L}{K}$ è algebrica così - come $\faktor{F}{L}$, allora anche - $\faktor{F}{K}$ è algebrica. + $\faktor{L}{K}$ è un'estensione finita se e solo se + è un'estensione finitamente generata da elementi algebrici. \end{proposition} - + \begin{proof} - Sia $f \in F$. Allora, poiché $F$ è - algebrico su $L$, esistono $l_0$, \ldots, - $l_n \in L$ tali per cui, detto - $p(x) = l_n x^n + \ldots + l_1 x + l_0 \in L[x]$, - vale che $p(f) = 0$. In particolare $f$ è - algebrico su $K(l_n, \ldots, l_0)$, e quindi - $K(l_n, \ldots, l_0, f)$ è un'estensione finita - su $K(l_n, \ldots, l_0)$. \medskip + Se $L$ è un'estensione finita su $K$, allora esiste una + base finita $\basis = \{l_1, \ldots, l_n\} \subseteq L$ + tale per cui $L = K(l_1, \ldots, l_n)$. Poiché $L$ è + un'estensione finita, $L$ è anche algebrica, e quindi + $\basis$ è composta da elementi algebrici su $K$. Pertanto + $L$ è un'estensione finitamente generata da elementi algebrici + su $K$. \medskip - - Chiaramente $K(l_n, \ldots, l_0)$ è un'estensione + Sia ora $L = K(l_1, \ldots, l_n)$ con $l_i$ elemento algebrico + su $K$. Allora, per il teorema delle torri algebriche, + $L$ è un'estensione finita su $K$ dal momento che questi due campi sono i due estremi della seguente torre di estensioni: \[\begin{tikzcd} @@ -289,21 +353,88 @@ \arrow[no head, from=3-1, to=4-1] \arrow[no head, from=4-1, to=5-1] \end{tikzcd}\] - Infatti ogni campo della torre è un'estensione + dove ogni campo interno della torre è un'estensione finita del sottocampo corrispondente dal momento - che $\faktor{L}{K}$ è un'estensione algebrica\footnote{ - In particolare questo dimostra che un'estensione - algebrica e finitamente generata è anche - finita. Si può generalizzare il risultato - mostrando che un'estensione è finita se e solo - se finitamente generata da elementi algebrici. - }. \medskip + che $\faktor{L}{K}$ è un'estensione algebrica, da + cui la tesi. + \end{proof} + + \begin{proposition} + Sia $K \subseteq L \subseteq F$ una torre di + estensioni. Allora $\faktor{F}{K}$ è un'estensione + algebrica se e solo se lo sono sia $\faktor{L}{K}$ + che $\faktor{F}{L}$. + \end{proposition} + + \begin{proof} + Se $\faktor{F}{K}$ è un'estensione algebrica, + a maggior ragione $\faktor{F}{L}$ è + algebrica, dal momento che ogni elemento $f \in K$ è + radice di un polinomio a coefficienti in $K$, e + quindi, in particolare, di un polinomio a coefficienti in + $L$. Allora stesso tempo, ogni elemento di $L$ è un + elemento di $F$, e quindi tale elemento è ancora algebrico + su $K$, e così anche $\faktor{L}{K}$ è un'estensione + algebrica. \medskip + + + Siano $\faktor{L}{K}$ e $\faktor{F}{L}$ + estensioni algebriche. Sia $f \in F$. Allora, poiché $F$ è + algebrico su $L$, esistono $l_0$, \ldots, + $l_n \in L$ tali per cui, detto + $p(x) = l_n x^n + \ldots + l_1 x + l_0 \in L[x]$, + vale che $p(f) = 0$. In particolare $f$ è + algebrico su $K(l_n, \ldots, l_0)$, e quindi + $K(l_n, \ldots, l_0, f)$ è un'estensione finita + su $K(l_n, \ldots, l_0)$. \medskip + + + Chiaramente anche $K(l_n, \ldots, l_0)$ è un'estensione + finita su $K$ dal momento che è finitamente generata + da elementi algebrici su $K$, dacché $L$ è un'estensione + algebrica su $K$. \medskip Per il teorema delle torri algebriche, allora $K(l_n, \ldots, l_0, f)$ è un'estensione finita - di $K$. Dal momento allora che $K(f) \subseteq K(l_n, \ldots, l_0, f)$, anche questa è un'estensione finita, - e quindi $f$ è algebrico, da cui la tesi. + su $K$. Dal momento allora che $K(f) \subseteq K(l_n, \ldots, l_0, f)$, anche questa è un'estensione finita, + e quindi $f$ è algebrico. Pertanto si conclude che + $\faktor{F}{K}$ è un'estensione algebrica, da cui + la tesi. + \end{proof} + + Infine, si presenta un risultato interessante che lega + l'algebricità di $\faktor{L}{K}$ e $\faktor{M}{K}$ a quella + di $\faktor{LM}{K}$: + + \begin{proposition} + Siano $K \subseteq L$, $M$. Allora le estensioni + $\faktor{L}{K}$ e $\faktor{M}{K}$ sono algebriche se + e solo se $\faktor{LM}{K}$ è algebrica. + \end{proposition} + + \begin{proof} + Siano $\faktor{L}{K}$ e $\faktor{M}{K}$ algebriche. + Sia $\alpha \in LM = L(M)$. Dal momento che $L(M)$ è un + $L$-spazio vettoriale i cui vettori sono gli elementi di + $M$, allora $\alpha$ può scriversi + come combinazione lineare finita di elementi in $M$ con + coefficienti in $L$, ossia: + \[ \alpha = \sum_{i=1}^n \lambda_i m_i. \] + Poiché $L$ e $M$ sono estensioni algebriche su $K$, + $K' := K(\lambda_1, \ldots, \lambda_n, m_1, \ldots, m_n)$ è + un'estensione finitamente generata da elementi algebrici + ed è pertanto finita su $K$. Poiché $K(\alpha) \subseteq + K'$, $K(\alpha)$ è un'estensione finita su $K$ e dunque + $\alpha$ è algebrico su $K$. Pertanto $LM$ è un'estensione + algebrica su $K$. \medskip + + + Se $\faktor{LM}{K}$ è un'estensione algebrica, allora + in particolare ogni elemento di $L$, che appartiene a $L$, + è algebrico su $K$, e così $\faktor{L}{K}$ è un'estensione + algebrica. Analogamente lo è anche $\faktor{M}{K}$, da cui + la tesi. \end{proof} \end{document} \ No newline at end of file