Add aluffi chapter 1

Author: jfuzaroa

Aluffi Algebra/Aluffi_Algebra.pdf

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+
+\documentclass[a4paper, 12pt]{report}
+
+\usepackage[top=36pt,bottom=36pt,left=48pt,right=48pt,marginparwidth=110pt,marginparsep=-80pt]{geometry}
+\usepackage{marginnote}
+\usepackage{enumitem}
+\usepackage{relsize}
+\usepackage{setspace}
+\usepackage{mathtools}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{commath}
+\usepackage{subfiles}
+
+% theorem, definition, etc. styles
+\newtheorem{thm}{Theorem}[chapter]
+\newtheorem{cor}[thm]{Corrollary}
+\newtheorem{lem}[thm]{Lemma}
+\newtheorem{prop}[thm]{Proposition}
+\newtheorem*{fact}{Fact}
+\theoremstyle{definition}
+\newtheorem{defn}[thm]{Definition}
+
+% commands
+\newcommand*{\N}{\mathbb{N}} % natural numbers
+\newcommand*{\Z}{\mathbb{Z}} % integers
+% \newcommand*{\gcd}[2]{\mathrm{gcd}(#1,#2)} % greatest common divisor
+
+\begin{document}
+
+\chapter{The Integers}
+\label{ch:intgs}
+
+\subfile{sections/1_the_integers.tex}
+
+\chapter{Modular Arithmetic}
+\label{ch:mod_arth}
+
+\subfile{sections/2_modular_arithmetic.tex}
+
+\end{document}

Aluffi Algebra/Aluffi_Algebra.tex

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+\documentclass[../Aluffi_Algebra.tex]{subfiles}
+
+\begin{document}
+
+\begin{defn}
+    Let \(a, b \in \Z\). We say that `\(b\) divides \(a\)', or `\(b\) is a divisor of \(a\)',
+    or `\(a\) is a multiple of \(b\)', and write \(b \mid a\), is an \emph{integer} \(c \in \Z\) such that \(a = bc\).
+\end{defn}
+
+\begin{lem}
+    If \(b \mid a\) and \(a \neq 0\), then \(\abs{b} \leq \abs{a}\).
+\end{lem}
+
+\begin{fact}[Well-ordering Principle]
+    Every nonempty set of nonnegative integers contains a least element.
+\end{fact}
+
+\begin{thm}[Division with remainder]
+    Let \(a, b \in \Z\) with \(b \neq 0\). Then there exists a unique `quotient' \(q \in \Z\) and a 
+    unique `remainder' \(r \in \Z\) such that
+    \[ a = bq + r \qquad\text{with } \abs{r} < \abs{b}. \]
+\end{thm}
+
+\stepcounter{thm}
+
+\begin{defn}
+    Let \(a,b \in \Z\). We say that a nonnegative integer \(d\) is the `greatest common divisor' of \(a\) and \(b\),
+    denoted \(\gcd(a,b)\) or simply \((a,b)\), if
+    \begin{itemize}
+        \item \(d \mid a\) and \(d \mid b\); and
+        \item if \(c \mid a\) and \(c \mid \)b, then \(c \mid d\).
+    \end{itemize}
+\end{defn}
+
+\stepcounter{thm}
+\stepcounter{thm}
+
+\begin{thm}
+    Let \(a, b \in \Z\). Then the greatest common divisor \(d = \gcd(a,b)\) is an integer linear combination of \(a\)
+    and \(b\). That is, there exists integers \(m\) and \(n\) such that \(d = ma + nb\).
+    \\ \\
+    In fact, if \(a\) and \(b\) are not both 0, then \(\gcd(a,b)\) is the smallest positive linear combination
+    of \(a\) and \(b\).
+\end{thm}
+
+\begin{cor}
+    Let \(a, b \in \Z\). Then \(\gcd(a,b) = 1\) if and only if \(1\) may be expressed as a linear combination of \(a\) and \(b\).
+\end{cor}
+
+\begin{defn}
+    We say that \(a\) and \(b\) are \emph{relatively prime} if \(\gcd(a,b) = 1\).
+\end{defn}
+
+\begin{cor}
+    Let \(a, b, c \in \Z\). If \(a \mid bc\) and \(\gcd(a,b) = 1\), then \(a \mid c\).
+\end{cor}
+
+\stepcounter{thm}
+\stepcounter{thm}
+
+\begin{thm}[Euclidean Algorithm]
+    Let \(a, b, \in \Z\), with \(b \neq 0\). Then with notation as above,
+    \(\gcd(a, b)\) equals the last nonzero remainder \(r_n\).\\
+    More explicitely: let \(r_{-2} = a\) and \(r_{-1} = b\); for \(i \geq 0\), let \(r_i\) be the remainder of the
+    division of \(r_{i-2}\) by \(r_{i-1}\). Then there is an integer \(n\) such
+    that \(r_n \neq 0\) and \(rn_{n+1} = 0\), and \(\gcd(a,b) = r_n\).
+\end{thm}
+
+\begin{lem}
+    Let \(a, b, q, r \in \Z\), with \(b \neq 0\), and assume that \(a = bq + r\). Then \(\gcd(a,b) = \gcd(b,r)\).
+\end{lem}
+
+\stepcounter{thm}
+
+\begin{defn}
+    An integer \(p\) is `irreducable' if \(p \neq \pm 1\) and the only divisors of \(p\) are \(\pm 1, \pm p\).\\
+    An integer \(\neq 0, \neq \pm 1\) is `reducible' or `composite' if it is not irreducable.
+\end{defn}
+
+\begin{lem}
+    Assume that \(p\) is an irreducible integer and that \(b\) is not a multiple of \(p\). Then \(b\) and \(p\)
+    are relatively prime, that is, \(\gcd(p,b) = 1\).
+\end{lem}
+
+\begin{defn}
+    An integer \(p\) is `prime' if \(p \neq \pm 1\) and whenver \(p\) divides the product \(bc\) of two integers
+    \(b,c\), then \(p \mid b\) or \(p \mid c\).
+\end{defn}
+
+\stepcounter{thm}
+
+\begin{thm}
+    Let \(p \in \Z, p \neq 0\). Then \(p\) is prime if and only if it is irreducable.
+\end{thm}
+
+\begin{thm}[Fundamental Theorem of Arithmetic]
+    Every integer \(n \neq 0, \neq \pm 1\) is a product of finitely many irredicubile integers: \(\forall n \in \Z, n
+    \neq 0, n \neq \pm 1\), there exists irreducible integers \(q_1, \ldots, q_r\) such that
+    \[ n = q_1 \cdots q_r, \]
+    \[ n = \prod_r q_r. \]
+    Further, this factorization is unique in the sense that if
+    \[ n = q_1 \cdots q_r = p_1 \cdots p_s, \]
+    with all \(q_i, p_j\) irreducible, then necessarily \(s = r\) and after reordering the factors we have 
+    \(p_1 = \pm q_1, p_2 = \pm q_2, \ldots, p_r = \pm q_r\).
+\end{thm}
+
+\stepcounter{thm}
+\stepcounter{thm}
+
+\begin{prop}
+    Let \(a, b \in \Z^{\neq 0}\), and write
+    \[a = \pm 2^{\alpha_2} 3^{\alpha_3} 5^{\alpha_5} 7^{\alpha_7} 11^{\alpha_{11} }\cdots,\]
+    \[b = \pm 2^{\beta_2}  3^{\beta_3}  5^{\beta_5}  7^{\beta_7}  11^{\beta_{11} }\cdots,\]
+    as above. Then the \(\gcd\) of \(a\) and \(b\) is the positive integer
+    \[d = 2^{\delta_2} 3^{\delta_3} 5^{\delta_5} 7^{\delta_7} 11^{\delta_11}\cdots,\]
+    where \(\delta_i = \min(\alpha_i,\beta_i)\) for all \(i\).
+\end{prop}
+
+\stepcounter{thm}
+
+\begin{cor}
+    Two nonzero integers \(a, b\) are relatively prime if and only if they have no common irreducible factor.
+\end{cor}
+
+\end{document}

Aluffi Algebra/sections/1_the_integers.tex

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+\documentclass[../Aluffi_Algebra.tex]{subfiles}
+
+\begin{document}
+
+\begin{defn}
+    Let \(n \geq 0\) be an integer, and let \(a, b \in \Z\). We say that `\(a\) is congruent to \(b\) modulo \(n\)',
+    denoted \(a \equiv b \mod n\), if \(b - a \in n\Z\).
+\end{defn}
+
+\end{document}