From b3585c11bb06bec5dc51ef2f86f9711ce1ebe79b Mon Sep 17 00:00:00 2001 From: Hearot Date: Fri, 13 Jan 2023 15:43:10 +0100 Subject: [PATCH] =?UTF-8?q?feat(algebrario):=20aggiunge=20la=20seminorma?= =?UTF-8?q?=20(e=20l'esempio=20di=20Z[=E2=88=9A10])?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .gitignore | 1 + Aritmetica/2. Anelli euclidei, PID e UFD.tex | 151 +++++++++++++++++++ Aritmetica/aritmetica.pdf | Bin 856322 -> 873898 bytes Aritmetica/aritmetica.tex | 1 + Aritmetica/evan.sty | 2 + 5 files changed, 155 insertions(+) diff --git a/.gitignore b/.gitignore index 694a4ce..740d2f4 100644 --- a/.gitignore +++ b/.gitignore @@ -3,6 +3,7 @@ *.fdb_latexmk *.fls *.synctex.gz +*.synctex(busy) *.toc *.out *.pre diff --git a/Aritmetica/2. Anelli euclidei, PID e UFD.tex b/Aritmetica/2. Anelli euclidei, PID e UFD.tex index 560103e..0c72279 100644 --- a/Aritmetica/2. Anelli euclidei, PID e UFD.tex +++ b/Aritmetica/2. Anelli euclidei, PID e UFD.tex @@ -643,3 +643,154 @@ euclidei. A/(p_1^{m_1}) \times \cdots \times A/(p_{n-1}^{m_{n-1}}) \times A/(p_n^{m_n}). \] \end{proof} + +\subsection{La seminorma di \texorpdfstring{$\ZZ[\sqrt{n}]$}{Z[√n]}} + +Si definisce innanzitutto $\ZZ[\sqrt{n}]$ nel seguente modo: + +\[ \ZZ[\sqrt{n}] = \{ a + b\sqrt{n} \mid a, b \in \ZZ \}. \] + +\begin{definition} + Si definisce \textbf{seminorma} di $\ZZ[\sqrt{n}]$ la seguente + funzione: + + \[ \ell : \ZZ[\sqrt{n}] \to \ZZ, \, a + b\sqrt{n} \mapsto a^2 - n b^2. \] +\end{definition} + +\begin{proposition} + La seminorma di $\ZZ[\sqrt{n}]$ è una funzione moltiplicativa. +\end{proposition} + +\begin{proof} + Dimostrare la tesi è equivalente al verificare la seguente identità: + + \[ (a^2-nb^2)(c^2-nd^2)=(ac+nbd)^2-n(ad+bc)^2, \] + + \vskip 0.1in + + come si verifica nelle seguenti righe: + + \begin{multline*} + (ac+nbd)^2-n(ad+bc)^2 = a^2c^2+n^2b^2d^2+\cancel{2acnbd}-na^2d^2-nb^2c^2-\cancel{2acnbd} = \\ + a^2(c^2-nd^2) -nb^2(c^2-nd^2) = (a^2-nb^2)(c^2-nd^2). + \end{multline*} +\end{proof} + +\begin{theorem} + \label{th:invertibili_z_sqrtn} + Un elemento $a \in \ZZ[\sqrt{n}]$ è invertibile se e solo se + $\ell(a) \in \{1, -1\}$. +\end{theorem} + +\begin{proof} + Si dimostrano le due implicazioni separatamente. \\ + + ($\implies$) \; Sia $a \in a \in \ZZ[\sqrt{n}]^*$. Allora esiste un + $b \in \ZZ[\sqrt{n}]^*$ tale che $ab = 1$. Applicando la seminorma + a entrambi i membri si ricava che: + + \[ \ell(ab) = 1 \implies \ell(a)\ell(b) = 1. \] + + \vskip 0.1in + + Gli unici invertibili di $\ZZ$ sono tuttavia $1$ e $-1$, + da cui la tesi. \\ + + ($\;\Longleftarrow\;$) \; Si consideri $a+b\sqrt{n} \in \ZZ[\sqrt{n}]$. + Sia $d = \ell(a) \in \{1, -1\}$ si ricava che: + + \[ a^2-nb^2 = d \implies (a+b\sqrt{n})(a-b\sqrt{n})=d \implies (a+b\sqrt{n})d\inv(a-b\sqrt{n})=1, \] + + \vskip 0.1in + + da cui la tesi. +\end{proof} + +\begin{example}[$\ZZsqrt{10}$ non è un UFD] + Il numero $6$ ammette due fattorizzazioni in irriducibili + completamente distinte in $\ZZsqrt{10}$. Dunque + $\ZZsqrt{10}$ non è un UFD. Conseguentemente non è né un anello + euclideo\footnote{Violerebbe altrimenti il \thref{th:euclidei_ufd}.}, né un + PID\footnote{Si usa ancora la proposizione, non dimostrata + in queste dispense, secondo cui un PID è sempre un UFD. Per + tale dimostrazione si rimanda ancora a \cite[pp.~124-126]{di2013algebra}.}. +\end{example} + +\begin{proof} + Dal momento che $6=16-10$, + possiamo fattorizzare $6$ come il prodotto + di $4+\sqrt{10}$ e $4-\sqrt{10}$. Tuttavia, dalla + fattorizzazione in $\ZZ$, sappiamo anche + che $6=2 \cdot 3$. \\ + + Dimostriamo che sia $2$ che $3$ sono irriducibili + in $\ZZ[\sqrt{10}]$. Se $2$ fosse riducibile, + si potrebbe scrivere come prodotto + di due fattori non invertibili: \\ + + \begin{equation} + \label{eq:es_z_sqrt10_fattorizzazione_2} + 2 = (a + b\sqrt{10})(c + d\sqrt{10}) \implies 4 = (a^2 - 10b^2)(c^2 - 10d^2). + \end{equation} + + \vskip 0.1in + + Poiché nessun fattore di $2$ è invertibile per ipotesi, per il + \thref{th:invertibili_z_sqrtn} nessuno dei due + fattori in \eqref{eq:es_z_sqrt10_fattorizzazione_2} può essere uguale a $1$ o $-1$. + Allora l'unica possibilità è che $a^2 - 10b^2$ sia uguale a $2$ o + $-2$. Se però così fosse, $a^2 \equiv \pm 2 \pod{10}$, che + non ammette soluzione. \\ + + Reiterando lo stesso ragionamento per $3$, + si ottiene $a^2 \equiv \pm 3 \pod{10}$, che anche + stavolta non ammette soluzione. Quindi sia $2$ che + $3$ sono irriducibili in $\ZZ[\sqrt{10}]$. \\ + + Analogamente dimostriamo che sia $4+\sqrt{10}$ che + $4-\sqrt{10}$ sono irriducibili. Si assuma che + $4+\sqrt{10}$ sia riducibile, allora si ricava che: + + \[ 4+\sqrt{10} = (a + b\sqrt{10})(c + d\sqrt{10}), \] + + \vskip 0.1in + + da cui, passando alle seminorme si ottiene che: + + \[ 6 = (a^2 - 10b^2)(c^2 - 10d^2). \] + + \vskip 0.1in + + Poiché entrambi i fattori sono non invertibili per ipotesi, + per il \thref{th:invertibili_z_sqrtn} ognuno di essi è + diverso da $1$ e $-1$, come visto prima. Quindi l'unica + possibilità è che $a^2 - 10b^2$ sia uguale a $\pm 2$ o + $\pm 3$. Tuttavia, da prima sappiamo che nessuna di queste + equazioni ammette soluzione. Quindi $4+\sqrt{10}$ è irriducibile, + e allo stesso modo si dimostra che anche $4-\sqrt{10}$ lo + è. \\ + + Ora si dimostra che $2$ non è associato né a $4 + \sqrt{10}$ + né a $4 - \sqrt{10}$. Se fossero associati, esisterebbe + un invertibile $a$ tale che $2 = (4 \pm \sqrt{10})a$. \\ + + Passando alle norme, si ricava che: + + \[ 4 = 6 \, \ell(a), \] + + \vskip 0.1in + + dove, ricordando che $\ell(a)=\pm 1$ per il \thref{th:invertibili_z_sqrtn}, + si ottiene: + + \[ 4 = \pm 6, \] + + \vskip 0.1in + + ossia un assurdo, \Lightning{}. \\ + + Poiché $2$ non è associato né a né a $4 + \sqrt{10}$ + né a $4 - \sqrt{10}$, le due fattorizzazioni sono due + fattorizzazioni in irriducibili completamente + distinte. Quindi $\ZZsqrt{10}$ non può essere un UFD. +\end{proof} diff --git a/Aritmetica/aritmetica.pdf b/Aritmetica/aritmetica.pdf index 26f4524e9d6c78b08158acc0653ccaf8f5c82f19..b35d2f9602b32fcdd2d2c6b6d8c2545c5dc7d6f7 100644 GIT binary patch delta 151186 zcmagGcOcc@|2W>gMt1flGP2!eTx9RejOtn$*`Z`54N^!bGekDY zYUp?F<@GMTUZ3Ch{&UasanAFc^L*^bEPs030m?wKg0S#*J88duM7h zhisNR+zKz&)EkquesJjKpn4$!BBoo@Q8F4Xs%HH+ms1xiClC=GO=@HJFD7uk&59 zL(C=Bhi(~FL@tNW7^y#?*s(6RA)$)bK0|Tq?f7Q(rt1rLjt6yy&Y^4zswo>g4(&O)(v*ksum)P$BbN5^h}6`Rbc# z#+c5<#Fy2?ij;KE+}KavaQ}2Q}9f@!^N9ZaF#0O!r3$Btua^?#ydvG zR*YlMDp9;h50{=v*UWO;RG?F!(~OLqd8rj?>GEUJh`s+Ha~WmzMb~}}nqLB+w6@w4 z_!ysMb3J)>uh$Wid2(#T%i&u$?nZ(9SLx&;!Qi~ZLJJ*6BCQuCPfkpITQ+Ozh}W&n zyip0aWf$#|J>J__*nieb@PkFb=6OXUBvU2hAw%$2`yoMDN_4O%>HJVf&JxQmNmDJrW_(i-eknpOpm@B z>edju)VQH}urT#Ji!j;q9}ZV%>-=Vx9v}MwqxG?TxfaBjubd|K^v9H}27A`Gt~@;eA)7%1^^FZ76E{yj<=(-dN{&jSYtch8+dBaR(VD zlM6&{Mz7?C4N{t$u*+j<IP1)UR~LNPI?7ov*drpR9u=0azs`(leey{ zef^=cU}eZ}yL9*a^G^?N_ 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zblcm1#PeN7*<$=I0&k`8dSv;vYrn@lu%|MGxR|5X!s_V#fZ-=-V@AS2(ql)=%5RAk ziz}(|X&f>HcA#L;|AR9?63*3#0mD>r1G>v<k8gA7#(@Ldt~`41$kE?Ba_I7H RmR`)OZTt59%BwHj_WzF&Z|ML4 diff --git a/Aritmetica/aritmetica.tex b/Aritmetica/aritmetica.tex index 183d891..04b04a5 100644 --- a/Aritmetica/aritmetica.tex +++ b/Aritmetica/aritmetica.tex @@ -10,6 +10,7 @@ \usepackage{amsopn} \usepackage{biblatex} \usepackage{bm} +\usepackage{cancel} \usepackage{csquotes} \usepackage{mathtools} \usepackage{marvosym} diff --git a/Aritmetica/evan.sty b/Aritmetica/evan.sty index 621b0af..3aaa4f5 100644 --- a/Aritmetica/evan.sty +++ b/Aritmetica/evan.sty @@ -94,6 +94,7 @@ \newcommand{\ZZi}{\mathbb{Z}[i]} \newcommand{\ZZom}{\mathbb{Z}[\omega]} \newcommand{\ZZpx}{\mathbb{Z}_p[x]} +\newcommand{\ZZsqrt}[1]{\mathbb{Z}[\sqrt{#1}]} \newcommand{\ZZx}{\mathbb{Z}[x]} \newcommand{\ii}{\mathbf{i}} @@ -152,6 +153,7 @@ \DeclareMathOperator{\ord}{ord} \newcommand{\defeq}{\overset{\mathrm{def}}{=}} +\newcommand{\defiff}{\overset{\mathrm{def}}{\iff}} % From the USAMO .tex files \newcommand{\dg}{^\circ}