From 68cf0b5d74d62d751f4b2cf77f00729925c4169f Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 15 Nov 2023 20:22:54 +0100 Subject: [PATCH] feat(algebra1): aggiunge la funziona lambda di Carmichael --- .../main.pdf | Bin 246861 -> 248828 bytes .../main.tex | 25 ++++++++++++++++++ 2 files changed, 25 insertions(+) diff --git a/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.pdf b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. 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Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.tex b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.tex index bc29c48..2f8cb17 100644 --- a/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.tex +++ b/Secondo anno/Algebra 1/Teoria dei gruppi/Teoremi fondamentali (Cauchy, isomorfismo, corrispondenza, struttura e Sylow)/4. Il teorema di struttura per gruppi abeliani finiti e decomposizione di U(Zn)/main.tex @@ -363,4 +363,29 @@ è ciclico, e quindi $2^k$ può essere solo $1$, $2$ o $4$. Si conclude così la dimostrazione del teorema. \end{proof} + + \begin{remark}[La funzione $\lambda(n)$ di Carmichael] + Si definisce la funzione $\lambda : \NN^+ \to \NN^+$ di Carmichael in + modo tale che $\lambda(n)$ sia il più piccolo intero positivo $m$ tale + per cui $a^m \equiv 1 \pod{n}$ per ogni $a$ coprimo con $n$. \medskip + + + Grazie al Teorema sulla decomposizione di $\ZZmulmod{n}$, calcolare + $\lambda(n)$ risulta piuttosto semplice. Infatti, $\lambda(n)$ è + esattamente il minimo comune multiplo di tutti gli ordini di + $\ZZmulmod{n}$. In particolare, $\lambda(n)$ divide sempre $\varphi(n)$ e + vale l'uguaglianza se e solo se esiste un elemento $x$ in $\ZZmulmod{n}$ di + ordine $\varphi(n)$, ossia se e solo se $\ZZmulmod{n}$ è ciclico (e dunque se + e solo se + $n$ è $1$, $2$, $4$, $p^k$ o $2p^k$ per $p$ primo dispari). + \end{remark} + + \begin{example}[$\lambda(1000)$] + Si calcola $\lambda(1000)$. Dal momento che $1000 = 2^3 \cdot 5^3$, vale + che $\ZZmulmod{1000} \cong \ZZmulmod{2^3} \times \ZZmulmod{5^3}$. + Dacché $5$ è dispari e $\varphi(5^3) = 5^3 - 5^2 = 100$, $\ZZmulmod{5^3} \cong \ZZmod{100}$, mentre + $\ZZmulmod{2^3} \cong \ZZmod{2} \times \ZZmod{2}$. Pertanto vale che: + \[ \ZZmulmod{1000} \cong \ZZmod{2} \times \ZZmod{2} \times \ZZmod{100}, \] + e quindi $\lambda(1000) = \mcm(2,2,100) = 100$. + \end{example} \end{document} \ No newline at end of file