From 8d563a931805e563f415719171a7b252a2ac298d Mon Sep 17 00:00:00 2001 From: Hearot Date: Sun, 21 Apr 2024 19:51:43 +0200 Subject: [PATCH] =?UTF-8?q?feat(algebra1):=20irriducibilit=C3=A0=20di=20Ph?= =?UTF-8?q?i=5Fn(x)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../main.pdf | Bin 283125 -> 314256 bytes .../main.tex | 30 ++++++++++++++++-- 2 files changed, 28 insertions(+), 2 deletions(-) diff --git a/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.pdf b/Secondo anno/Algebra 1/3. 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Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex index a86042f..14f15c1 100644 --- a/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex +++ b/Secondo anno/Algebra 1/3. Teoria delle estensioni di campo e di Galois/Scheda riassuntiva di Teoria dei campi e di Galois/main.tex @@ -409,14 +409,14 @@ $a_1$, ..., $a_n$ sono le radici di $p$, $\Gal(L / K)$ agisce su $\{a_1, \ldots, a_n\}$ mediante $\Xi$, in modo tale che: - \vskip -0.3in + \begin{equation*} \begin{split} \Xi : \Gal(&L / K) \to S(\{a_1, \ldots, a_n\}) \cong S_n, \\ &\varphi_i \xmapsto{\Xi} [a_j \mapsto \varphi_i(a_j)]. \end{split} \end{equation*} - \vskip -0.2in + In particolare tale azione è transitiva (dunque $\Orb(a_i) = \{a_j\}_{j=1-n}$)e fedele. Poiché $\Xi$ è fedele, vale che $\Gal(L / K) \mono S_n$. Se $\Gal(L / K)$ è abeliano (e in tal caso si dice che $L$ è un'\textbf{estensione abeliana}), $\Xi$ è anche transitiva, e quindi @@ -638,6 +638,32 @@ \end{cases} \] + \subsection{Radici di primi in $\QQ$} + + Siano $p_1$, ..., $p_n$ numeri primi distinti. + Allora + vale che: + \[ \Gal(\QQ(\sqrt{p_1}, \ldots, \sqrt{p_n})/\QQ) \cong (\ZZmod{2})^n. \] + + \subsection{I polinomi ciclotomici $\Phi_n(x)$} + + Sia $\Phi_n(x)$ l'$n$-esimo polinomio ciclotomico, così definito: + \[ \Phi_n(x) = \prod_{\substack{1 \leq d \leq n \\ \MCD(d, n) = 1}} (x - \zeta_n^d), \] + dove $\zeta_n$ è una radice primitiva $n$-esima dell'unità. \medskip + + + Tale polinomio è sempre a coefficienti interi ed è inoltre primitivo + su $\ZZ[x]$. Vale inoltre che: + \[ x^n - 1 = \prod_{m \mid n} \Phi_m(x). \] + Il campo di spezzamento di $\Phi_n(x)$ su $\QQ$ è + $\QQ(\zeta_n)$, che è un'estensione normale, separabile e finita, + e pertanto di Galois. \medskip + + + Inoltre vale che: + \[ \Gal(\QQ(\zeta_n)/\QQ) \cong (\ZZmod{n})^\times, \] + e dunque $\Phi_n(x)$ è sempre irriducibile su $\QQ$. + \vfill \hrule ~\\