mirror of https://github.com/hearot/notes
chore: divide i vecchi capitoli dell'Algebrario in file distinti
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\chapter*{Premessa}
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TODO
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\chapter{Teorema fondamentale dell'Algebra e radici reali in \texorpdfstring{$\QQx$}{Q[x]}}
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Si enuncia adesso il \nameref{th:algebra}, senza tuttavia
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fornirne una dimostrazione\footnote{Per la dimostrazione si rimanda
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a \cite[pp.~142-143]{di2013algebra}, avvisando della sua
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estrema tecnicità. Una dimostrazione a tema strettamente
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algebrico è dovuta invece al matematico francese Laplace (1749 -- 1827), per la quale
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si rimanda a \cite[pp.~120-122]{Remmert1991}.}.
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\begin{theorem}[\textit{Teorema fondamentale dell'Algebra}]
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\label{th:algebra}
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Un polinomio non costante $f(x) \in \CCx$ ammette sempre almeno una radice in
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$\CC$.
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\end{theorem}
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\begin{corollary}
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Sia $f(x) \in \CCx$ di grado $n\geq1$. Allora $f(x)$ ammette
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esattamente $n$ radici, contate con la giusta molteplicità.
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\end{corollary}
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\begin{proof}
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Sia $\zeta_1$ una radice complessa di $f(x)$, la cui esistenza
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è garantita dal \nameref{th:algebra}. Si divida $f(x)$ per
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$(x-\zeta_1)$ e se ne prende il quoziente $q_1(x)$, mentre si
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ignori il resto, che
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per la \textit{Proposizione \ref{prop:radice_x_meno_alpha}},
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è nullo. \\
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Si reiteri il procedimento utilizzando $q_1(x)$ al
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posto di $f(x)$ fino a quando il grado del quoziente non è nullo,
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e si chiami infine questo quoziente di grado nullo $\alpha$.
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Infatti, poiché i gradi dei quozienti diminuiscono di $1$ ad
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ogni iterazione, è garantito che l'algoritmo termini esattamente
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dopo $n$ iterazioni. Pertanto, $f(x)$ a priori ha almeno $n$ radici. \\
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In questo modo, numerando le radici, si può scrivere $f(x)$ come:
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\begin{equation}
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\label{eq:fattorizzazione_fx__reali}
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f(x)=\alpha(x-\zeta_1)(x-\zeta_2)\cdots(x-\zeta_n).
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\end{equation}
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\vskip 0.1in
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Dal momento che $x-\zeta_i$ è irriducibile $\forall 1 \leq i \leq n$
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e dacché $\KKx$, in quanto anello euclideo, è un UFD, si dimostra
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che \eqref{eq:fattorizzazione_fx__reali} è l'unica fattorizzazione di
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$f(x)$, a meno di associati. Pertanto $f(x)$ ammette esattamente
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$n$ radici.
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\end{proof}
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\PassOptionsToPackage{main=italian}{babel}
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\documentclass[11pt]{scrbook}
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\usepackage{evan}
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\usepackage[utf8]{inputenc}
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\usepackage[italian]{babel}
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\usepackage{algorithm2e}
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\usepackage{amsfonts}
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\usepackage{amsthm}
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\usepackage{amssymb}
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\usepackage{amsopn}
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\usepackage{biblatex}
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\usepackage{bm}
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\usepackage{cancel}
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\usepackage{csquotes}
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\usepackage{mathtools}
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\usepackage{marvosym}
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\usepackage{tikz}
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\usepackage{wrapfig}
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\usepackage{xpatch}
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\addbibresource{bibliography.bib}
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\begin{document}
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\title{L'Algebrario}
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\subtitle{dispense del corso di Aritmetica}
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\author{Gabriel Antonio Videtta}
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\date{A.A. 2022/2023 \\ \vskip 1in \includegraphics[scale=0.3]{logo.png}}
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\maketitle
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{0. Premessa}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\tableofcontents
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{1. Introduzione alla teoria degli anelli}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{2. Anelli euclidei, PID e UFD}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{3. Esempi notevoli di anelli euclidei}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{4. Proprietà fondamentali di Z[i], Zp[x], Z[x], Q[x]}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{5. Irriducibilità in Z[x] e Q[x]}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{6. Proprietà dei polinomi di K[x] e delle estensioni algebriche}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{7. Estensioni algebriche di K}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{8. Teorema fondamentale dell'algebra ed estensioni di Q}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{9. Introduzione a teoria dei campi}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{10. Teoremi rilevanti sui campi finiti}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\include{11. Polinomi simmetrici e Teorema fondamentale dell'Algebra}
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\newpage
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\thispagestyle{empty}
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~\newpage
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\chapter{Riferimenti bibliografici}
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\printbibliography[heading=none]
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\end{document}
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@book{di2013algebra,
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title={Algebra},
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author={Di Martino, P. and Dvornicich, R.},
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isbn={9788867410958},
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series={Didattica e Ricerca. Manuali},
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year={2013},
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publisher={Pisa University Press},
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shorthand={DM}
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}
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@book{herstein2010algebra,
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title={Algebra},
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author={Herstein, I.N.},
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isbn={9788864732107},
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year={2010},
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publisher={Editori Riuniti University Press},
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shorthand={H}
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}
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@Inbook{Remmert1991,
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author="Remmert, R.",
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title="The Fundamental Theorem of Algebra",
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bookTitle="Numbers",
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year="1991",
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publisher="Springer New York",
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address="New York, NY",
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pages="97--122",
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isbn="978-1-4612-1005-4",
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doi="10.1007/978-1-4612-1005-4_5",
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url="https://doi.org/10.1007/978-1-4612-1005-4_5"
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}
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@article{10.2307/2315810,
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ISSN = {00029890, 19300972},
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URL = {http://www.jstor.org/stable/2315810},
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author = {M. A. Jodeit},
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journal = {The American Mathematical Monthly},
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number = {7},
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pages = {835--836},
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publisher = {Mathematical Association of America},
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title = {Uniqueness in the Division Algorithm},
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volume = {74},
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year = {1967}
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}
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@book{di2013algebra,
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title={Algebra},
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author={Di Martino, P. and Dvornicich, R.},
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isbn={9788867410958},
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series={Didattica e Ricerca. Manuali},
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year={2013},
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publisher={Pisa University Press},
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shorthand={DM}
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}
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@article{10.2307/2315810,
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ISSN = {00029890, 19300972},
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URL = {http://www.jstor.org/stable/2315810},
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author = {M. A. Jodeit},
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journal = {The American Mathematical Monthly},
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number = {7},
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pages = {835--836},
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publisher = {Mathematical Association of America},
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title = {Uniqueness in the Division Algorithm},
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volume = {74},
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year = {1967}
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}
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@book{herstein2010algebra,
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title={Algebra},
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author={Herstein, I.N.},
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isbn={9788864732107},
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year={2010},
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publisher={Editori Riuniti University Press},
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shorthand={H}
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% | \ %
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% \▓▓▓▓▓▓▓ \▓ \▓▓▓▓▓▓▓\▓▓ \▓▓\▓▓\▓▓▓▓▓▓▓ \▓▓▓▓ _\▓▓▓▓▓▓▓ %
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% \▓▓▓▓▓▓ %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% https://github.com/vEnhance/dotfiles/blob/main/texmf/tex/latex/evan/evan.sty
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% TL;DR of the Boost license conditions are as follows:
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%
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% 1. Any SOURCE VERSIONS must cite evan.sty and the Boost license below.
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% 2. For COMPILED PDF OUTPUT, attribution of evan.sty is OPTIONAL (but nice).
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% BOOST SOFTWARE LICENSE - VERSION 1.0 - 17 AUGUST 2003
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%
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% Copyright (c) 2022 Evan Chen [evan at evanchen.cc]
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% https://web.evanchen.cc/ || github.com/vEnhance
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%
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% https://github.com/vEnhance/dotfiles/blob/main/texmf/tex/latex/evan/evan.sty
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%
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% Permission is hereby granted, free of charge, to any person or organization
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% obtaining a copy of the software and accompanying documentation covered by
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% this license (the "Software") to use, reproduce, display, distribute,
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% execute, and transmit the Software, and to prepare derivative works of the
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% Software, and to permit third-parties to whom the Software is furnished to
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% do so, all subject to the following:
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%
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% The copyright notices in the Software and this entire statement, including
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% the above license grant, this restriction and the following disclaimer,
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% must be included in all copies of the Software, in whole or in part, and
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% all derivative works of the Software, unless such copies or derivative
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% works are solely in the form of machine-executable object code generated by
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% IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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% FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT
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% SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE
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\newcommand{\FFpn}{\mathbb{F}_{p^n}}
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\newcommand{\FFp}[1]{\mathbb{F}_{p^{#1}}}
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% From H113 "Introduction to Abstract Algebra" at UC Berkeley
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\let\circ\undefined
|
||||
\DeclareMathOperator{\circ}{\oldcirc}
|
||||
|
||||
\let\oldexists\exists
|
||||
\let\exists\undefined
|
||||
\DeclareMathOperator{\exists}{\oldexists}
|
||||
|
||||
\let\oldforall\forall
|
||||
\let\forall\undefined
|
||||
\DeclareMathOperator{\forall}{\oldforall}
|
||||
|
||||
\let\oldnexists\nexists
|
||||
\let\nexists\undefined
|
||||
\DeclareMathOperator{\nexists}{\oldnexists}
|
||||
|
||||
\let\oldland\land
|
||||
\let\land\undefined
|
||||
\DeclareMathOperator{\land}{\oldland}
|
||||
|
||||
\let\oldlnot\lnot
|
||||
\let\lnot\undefined
|
||||
\DeclareMathOperator{\lnot}{\oldlnot}
|
||||
|
||||
\let\oldlor\lor
|
||||
\let\lor\undefined
|
||||
\DeclareMathOperator{\lor}{\oldlor}
|
||||
|
||||
\title{}
|
||||
\author{Gabriel Antonio Videtta}
|
||||
\date{\today}
|
||||
|
||||
\PassOptionsToPackage{usenames,svgnames,dvipsnames,table}{xcolor}
|
||||
\usepackage{xcolor}
|
||||
\usepackage[colorlinks=true]{hyperref}
|
||||
\hypersetup{urlcolor=RubineRed,linkcolor=RoyalBlue,citecolor=ForestGreen}
|
||||
\usepackage[nameinlink]{cleveref}
|
||||
|
||||
\usepackage{amsthm}
|
||||
\usepackage{thmtools}
|
||||
\usepackage[framemethod=TikZ]{mdframed}
|
||||
|
||||
\mdfdefinestyle{mdbluebox}{%
|
||||
roundcorner=10pt,
|
||||
linewidth=1pt,
|
||||
skipabove=12pt,
|
||||
innerbottommargin=9pt,
|
||||
skipbelow=2pt,
|
||||
linecolor=blue,
|
||||
nobreak=true,
|
||||
backgroundcolor=TealBlue!5,
|
||||
}
|
||||
\declaretheoremstyle[
|
||||
headfont=\sffamily\bfseries\color{MidnightBlue},
|
||||
mdframed={style=mdbluebox},
|
||||
headpunct={\\[3pt]},
|
||||
postheadspace={0pt}
|
||||
]{thmbluebox}
|
||||
|
||||
\mdfdefinestyle{mdbluebox2}{%
|
||||
roundcorner=10pt,
|
||||
linewidth=1pt,
|
||||
skipabove=12pt,
|
||||
innerbottommargin=9pt,
|
||||
skipbelow=2pt,
|
||||
linecolor=blue,
|
||||
nobreak=true,
|
||||
backgroundcolor=BlueViolet!9,
|
||||
}
|
||||
\declaretheoremstyle[
|
||||
headfont=\sffamily\bfseries\color{RoyalPurple},
|
||||
mdframed={style=mdbluebox2},
|
||||
headpunct={\\[3pt]},
|
||||
postheadspace={0pt}
|
||||
]{thmbluebox2}
|
||||
|
||||
\mdfdefinestyle{mdredbox}{%
|
||||
linewidth=0.5pt,
|
||||
skipabove=12pt,
|
||||
frametitleaboveskip=5pt,
|
||||
frametitlebelowskip=0pt,
|
||||
skipbelow=2pt,
|
||||
frametitlefont=\bfseries,
|
||||
innertopmargin=4pt,
|
||||
innerbottommargin=8pt,
|
||||
nobreak=true,
|
||||
backgroundcolor=Salmon!5,
|
||||
linecolor=RawSienna,
|
||||
}
|
||||
\declaretheoremstyle[
|
||||
headfont=\bfseries\color{RawSienna},
|
||||
mdframed={style=mdredbox},
|
||||
headpunct={\\[3pt]},
|
||||
postheadspace={0pt},
|
||||
]{thmredbox}
|
||||
|
||||
\mdfdefinestyle{mdredbox2}{%
|
||||
roundcorner=10pt,
|
||||
linewidth=1pt,
|
||||
skipabove=12pt,
|
||||
innerbottommargin=9pt,
|
||||
skipbelow=2pt,
|
||||
linecolor=red,
|
||||
nobreak=true,
|
||||
backgroundcolor=WildStrawberry!5,
|
||||
}
|
||||
\declaretheoremstyle[
|
||||
headfont=\sffamily\bfseries\color{Maroon},
|
||||
mdframed={style=mdredbox2},
|
||||
headpunct={\\[3pt]},
|
||||
postheadspace={0pt}
|
||||
]{thmredbox2}
|
||||
|
||||
\mdfdefinestyle{mdgreenbox}{%
|
||||
skipabove=8pt,
|
||||
linewidth=2pt,
|
||||
rightline=false,
|
||||
leftline=true,
|
||||
topline=false,
|
||||
bottomline=false,
|
||||
linecolor=ForestGreen,
|
||||
backgroundcolor=ForestGreen!5,
|
||||
}
|
||||
\declaretheoremstyle[
|
||||
headfont=\bfseries\sffamily\color{ForestGreen!70!black},
|
||||
bodyfont=\normalfont,
|
||||
spaceabove=2pt,
|
||||
spacebelow=1pt,
|
||||
mdframed={style=mdgreenbox},
|
||||
headpunct={ --- },
|
||||
]{thmgreenbox}
|
||||
|
||||
\mdfdefinestyle{mdgreenbox2}{%
|
||||
roundcorner=10pt,
|
||||
linewidth=1pt,
|
||||
skipabove=12pt,
|
||||
innerbottommargin=9pt,
|
||||
skipbelow=2pt,
|
||||
linecolor=ForestGreen,
|
||||
nobreak=true,
|
||||
backgroundcolor=ForestGreen!5,
|
||||
}
|
||||
\declaretheoremstyle[
|
||||
headfont=\sffamily\bfseries\color{ForestGreen!70!black},
|
||||
mdframed={style=mdgreenbox2},
|
||||
headpunct={\\[3pt]},
|
||||
postheadspace={0pt}
|
||||
]{thmgreenbox2}
|
||||
|
||||
\mdfdefinestyle{mdblackbox}{%
|
||||
skipabove=8pt,
|
||||
linewidth=3pt,
|
||||
rightline=false,
|
||||
leftline=true,
|
||||
topline=false,
|
||||
bottomline=false,
|
||||
linecolor=black,
|
||||
backgroundcolor=RedViolet!5!gray!10,
|
||||
}
|
||||
\declaretheoremstyle[
|
||||
headfont=\bfseries,
|
||||
bodyfont=\normalfont\small,
|
||||
spaceabove=0pt,
|
||||
spacebelow=0pt,
|
||||
mdframed={style=mdblackbox}
|
||||
]{thmblackbox}
|
||||
|
||||
\mdfdefinestyle{mdblackbox2}{%
|
||||
skipabove=8pt,
|
||||
linewidth=3pt,
|
||||
rightline=false,
|
||||
leftline=true,
|
||||
topline=false,
|
||||
bottomline=false,
|
||||
linecolor=gray,
|
||||
backgroundcolor=RedViolet!5!gray!10,
|
||||
}
|
||||
\declaretheoremstyle[
|
||||
headfont=\bfseries,
|
||||
bodyfont=\normalfont\small,
|
||||
spaceabove=0pt,
|
||||
spacebelow=0pt,
|
||||
mdframed={style=mdblackbox2}
|
||||
]{thmblackbox2}
|
||||
|
||||
\newcommand{\listhack}{$\empty$\vspace{-2em}}
|
||||
|
||||
\declaretheorem[%
|
||||
style=thmbluebox,name=Teorema,numberwithin=section]{theorem}
|
||||
|
||||
\declaretheorem[style=thmgreenbox2,name=Lemma,sibling=theorem]{lemma}
|
||||
\declaretheorem[style=thmredbox2,name=Proposizione,sibling=theorem]{proposition}
|
||||
\declaretheorem[style=thmbluebox2,name=Corollario,sibling=theorem]{corollary}
|
||||
\declaretheorem[style=thmbluebox,name=Teorema,numbered=no]{theorem*}
|
||||
\declaretheorem[style=thmgreenbox2,name=Lemma,numbered=no]{lemma*}
|
||||
\declaretheorem[style=thmredbox2,name=Proposizione,numbered=no]{proposition*}
|
||||
\declaretheorem[style=thmbluebox2,name=Corollario,numbered=no]{corollary*}
|
||||
\declaretheorem[style=thmgreenbox,name=Claim,sibling=theorem]{claim}
|
||||
\declaretheorem[style=thmgreenbox,name=Claim,numbered=no]{claim*}
|
||||
\declaretheorem[style=thmredbox,name=Esempio,sibling=theorem]{example}
|
||||
\declaretheorem[style=thmredbox,name=Esempio,numbered=no]{example*}
|
||||
\declaretheorem[style=thmblackbox, name=Definizione,sibling=theorem]{definition}
|
||||
\declaretheorem[style=thmblackbox, name=Definizione,numbered=no]{definition*}
|
||||
\declaretheorem[style=thmblackbox2,name=Osservazione,sibling=theorem]{remark}
|
||||
\declaretheorem[style=thmblackbox2,name=Osservazione,numbered=no]{remark*}
|
||||
\declaretheorem[name=Congettura,sibling=theorem]{conjecture}
|
||||
\declaretheorem[name=Congettura,numbered=no]{conjecture*}
|
||||
\declaretheorem[name=Esercizio,sibling=theorem]{exercise}
|
||||
\declaretheorem[name=Esercizio,numbered=no]{exercise*}
|
||||
\declaretheorem[name=Asserzione,sibling=theorem]{fact}
|
||||
\declaretheorem[name=Asserzione,numbered=no]{fact*}
|
||||
\declaretheorem[name=Problema,sibling=theorem]{problem}
|
||||
\declaretheorem[name=Problema,numbered=no]{problem*}
|
||||
\declaretheorem[name=Domanda,sibling=theorem]{ques}
|
||||
\declaretheorem[name=Domanda,numbered=no]{ques*}
|
||||
|
||||
\Crefname{claim}{Claim}{Claim}
|
||||
\Crefname{conjecture}{Congettura}{Congetture}
|
||||
\Crefname{exercise}{Esercizio}{Esercizi}
|
||||
\Crefname{fact}{Asserzione}{Asserzioni}
|
||||
\Crefname{problem}{Problema}{Problemi}
|
||||
\Crefname{ques}{Domanda}{Domande}
|
||||
|
||||
\addtokomafont{partprefix}{}
|
||||
\renewcommand*{\partformat}{}
|
||||
|
||||
\@ifundefined{chapter}{}{
|
||||
\addtokomafont{partprefix}{}
|
||||
\renewcommand*{\partformat}{}
|
||||
\addtokomafont{chapterprefix}{}
|
||||
\RedeclareSectionCommand{chapter}
|
||||
\renewcommand*{\chapterformat}{}
|
||||
}
|
||||
|
||||
\renewcommand*{\sectionformat}%
|
||||
{\color{purple}\S\thesection\enskip}
|
||||
\renewcommand*{\subsectionformat}%
|
||||
{\color{purple}\S\thesubsection\enskip}
|
||||
\renewcommand*{\subsubsectionformat}%
|
||||
{\color{purple}\S\thesubsubsection\enskip}
|
||||
\KOMAoptions{numbers=noenddot}
|
||||
|
||||
\usepackage{listings}
|
||||
\usepackage{mathrsfs}
|
||||
\usepackage{textcomp}
|
||||
\lstset{basicstyle=\ttfamily\scriptsize,
|
||||
backgroundcolor=\color{green!2!white},
|
||||
breakatwhitespace=true,
|
||||
breaklines=true,
|
||||
commentstyle=\color{green!70!black},
|
||||
frame=shadowbox,
|
||||
frame=single,
|
||||
identifierstyle=\color{green!20!black},
|
||||
keywordstyle=\bfseries,
|
||||
keywordstyle=\bfseries\color{blue},
|
||||
numbers=left,
|
||||
numbersep=5pt,
|
||||
numberstyle=\tiny\sffamily\itshape\color{black!60},
|
||||
rulecolor=\color{blue!70!black},
|
||||
rulesepcolor=\color{blue!30!black},
|
||||
showstringspaces=false,
|
||||
stringstyle=\color{orange},
|
||||
tabsize=4,
|
||||
}
|
||||
|
||||
\usepackage[shortlabels]{enumitem}
|
||||
\usepackage[obeyFinal,textsize=scriptsize,shadow]{todonotes}
|
||||
\usepackage{textcomp}
|
||||
\usepackage{multirow}
|
||||
\usepackage{ellipsis}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{microtype}
|
||||
\usepackage{xstring}
|
||||
\usepackage{wrapfig}
|
||||
|
||||
\addtokomafont{subtitle}{\Large}
|
||||
\setkomafont{author}{\Large\scshape}
|
||||
\setkomafont{date}{\Large\normalsize}
|
||||
|
||||
\providecommand{\thetitle}{\@title}
|
||||
\providecommand{\theauthor}{\@author}
|
||||
\providecommand{\thedate}{\@date}
|
||||
|
||||
\usepackage[headsepline]{scrlayer-scrpage}
|
||||
\renewcommand{\headfont}{}
|
||||
\addtolength{\textheight}{3.14cm}
|
||||
\setlength{\footskip}{0.5in}
|
||||
\setlength{\headsep}{10pt}
|
||||
|
||||
\ihead{\footnotesize\textbf{\rightmark}}
|
||||
\ohead{\footnotesize Gabriel Antonio Videtta}
|
||||
|
||||
\automark{section}
|
||||
\chead{}
|
||||
\cfoot{\pagemark}
|
||||
|
||||
\hfuzz=\maxdimen
|
||||
\tolerance=10000
|
||||
\hbadness=10000
|
||||
|
||||
\KOMAoptions{twoside=false}
|
||||
|
Before Width: | Height: | Size: 93 KiB After Width: | Height: | Size: 93 KiB |
Loading…
Reference in New Issue