Created the Abstract Algebra theorems and definitions cheat sheet
This commit is contained in:
+303
@@ -0,0 +1,303 @@
|
||||
## Core latex/pdflatex auxiliary files:
|
||||
*.aux
|
||||
*.lof
|
||||
*.log
|
||||
*.lot
|
||||
*.fls
|
||||
*.out
|
||||
*.toc
|
||||
*.fmt
|
||||
*.fot
|
||||
*.cb
|
||||
*.cb2
|
||||
.*.lb
|
||||
|
||||
## Intermediate documents:
|
||||
*.dvi
|
||||
*.xdv
|
||||
*-converted-to.*
|
||||
# these rules might exclude image files for figures etc.
|
||||
# *.ps
|
||||
# *.eps
|
||||
# *.pdf
|
||||
|
||||
## Bibliography auxiliary files (bibtex/biblatex/biber):
|
||||
*.bbl
|
||||
*.bcf
|
||||
*.blg
|
||||
*-blx.aux
|
||||
*-blx.bib
|
||||
*.run.xml
|
||||
|
||||
## Build tool auxiliary files:
|
||||
*.fdb_latexmk
|
||||
*.synctex
|
||||
*.synctex(busy)
|
||||
*.synctex.gz
|
||||
*.synctex.gz(busy)
|
||||
*.pdfsync
|
||||
|
||||
## Build tool directories for auxiliary files
|
||||
# latexrun
|
||||
latex.out/
|
||||
|
||||
## Auxiliary and intermediate files from other packages:
|
||||
# algorithms
|
||||
*.alg
|
||||
*.loa
|
||||
|
||||
# achemso
|
||||
acs-*.bib
|
||||
|
||||
# amsthm
|
||||
*.thm
|
||||
|
||||
# beamer
|
||||
*.nav
|
||||
*.pre
|
||||
*.snm
|
||||
*.vrb
|
||||
|
||||
# changes
|
||||
*.soc
|
||||
|
||||
# comment
|
||||
*.cut
|
||||
|
||||
# cprotect
|
||||
*.cpt
|
||||
|
||||
# elsarticle (documentclass of Elsevier journals)
|
||||
*.spl
|
||||
|
||||
# endnotes
|
||||
*.ent
|
||||
|
||||
# fixme
|
||||
*.lox
|
||||
|
||||
# feynmf/feynmp
|
||||
*.mf
|
||||
*.mp
|
||||
*.t[1-9]
|
||||
*.t[1-9][0-9]
|
||||
*.tfm
|
||||
|
||||
#(r)(e)ledmac/(r)(e)ledpar
|
||||
*.end
|
||||
*.?end
|
||||
*.[1-9]
|
||||
*.[1-9][0-9]
|
||||
*.[1-9][0-9][0-9]
|
||||
*.[1-9]R
|
||||
*.[1-9][0-9]R
|
||||
*.[1-9][0-9][0-9]R
|
||||
*.eledsec[1-9]
|
||||
*.eledsec[1-9]R
|
||||
*.eledsec[1-9][0-9]
|
||||
*.eledsec[1-9][0-9]R
|
||||
*.eledsec[1-9][0-9][0-9]
|
||||
*.eledsec[1-9][0-9][0-9]R
|
||||
|
||||
# glossaries
|
||||
*.acn
|
||||
*.acr
|
||||
*.glg
|
||||
*.glo
|
||||
*.gls
|
||||
*.glsdefs
|
||||
*.lzo
|
||||
*.lzs
|
||||
*.slg
|
||||
*.slo
|
||||
*.sls
|
||||
|
||||
# uncomment this for glossaries-extra (will ignore makeindex's style files!)
|
||||
# *.ist
|
||||
|
||||
# gnuplot
|
||||
*.gnuplot
|
||||
*.table
|
||||
|
||||
# gnuplottex
|
||||
*-gnuplottex-*
|
||||
|
||||
# gregoriotex
|
||||
*.gaux
|
||||
*.glog
|
||||
*.gtex
|
||||
|
||||
# htlatex
|
||||
*.4ct
|
||||
*.4tc
|
||||
*.idv
|
||||
*.lg
|
||||
*.trc
|
||||
*.xref
|
||||
|
||||
# hyperref
|
||||
*.brf
|
||||
|
||||
# knitr
|
||||
*-concordance.tex
|
||||
# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
|
||||
# *.tikz
|
||||
*-tikzDictionary
|
||||
|
||||
# listings
|
||||
*.lol
|
||||
|
||||
# luatexja-ruby
|
||||
*.ltjruby
|
||||
|
||||
# makeidx
|
||||
*.idx
|
||||
*.ilg
|
||||
*.ind
|
||||
|
||||
# minitoc
|
||||
*.maf
|
||||
*.mlf
|
||||
*.mlt
|
||||
*.mtc[0-9]*
|
||||
*.slf[0-9]*
|
||||
*.slt[0-9]*
|
||||
*.stc[0-9]*
|
||||
|
||||
# minted
|
||||
_minted*
|
||||
*.pyg
|
||||
|
||||
# morewrites
|
||||
*.mw
|
||||
|
||||
# newpax
|
||||
*.newpax
|
||||
|
||||
# nomencl
|
||||
*.nlg
|
||||
*.nlo
|
||||
*.nls
|
||||
|
||||
# pax
|
||||
*.pax
|
||||
|
||||
# pdfpcnotes
|
||||
*.pdfpc
|
||||
|
||||
# sagetex
|
||||
*.sagetex.sage
|
||||
*.sagetex.py
|
||||
*.sagetex.scmd
|
||||
|
||||
# scrwfile
|
||||
*.wrt
|
||||
|
||||
# svg
|
||||
svg-inkscape/
|
||||
|
||||
# sympy
|
||||
*.sout
|
||||
*.sympy
|
||||
sympy-plots-for-*.tex/
|
||||
|
||||
# pdfcomment
|
||||
*.upa
|
||||
*.upb
|
||||
|
||||
# pythontex
|
||||
*.pytxcode
|
||||
pythontex-files-*/
|
||||
|
||||
# tcolorbox
|
||||
*.listing
|
||||
|
||||
# thmtools
|
||||
*.loe
|
||||
|
||||
# TikZ & PGF
|
||||
*.dpth
|
||||
*.md5
|
||||
*.auxlock
|
||||
|
||||
# titletoc
|
||||
*.ptc
|
||||
|
||||
# todonotes
|
||||
*.tdo
|
||||
|
||||
# vhistory
|
||||
*.hst
|
||||
*.ver
|
||||
|
||||
# easy-todo
|
||||
*.lod
|
||||
|
||||
# xcolor
|
||||
*.xcp
|
||||
|
||||
# xmpincl
|
||||
*.xmpi
|
||||
|
||||
# xindy
|
||||
*.xdy
|
||||
|
||||
# xypic precompiled matrices and outlines
|
||||
*.xyc
|
||||
*.xyd
|
||||
|
||||
# endfloat
|
||||
*.ttt
|
||||
*.fff
|
||||
|
||||
# Latexian
|
||||
TSWLatexianTemp*
|
||||
|
||||
## Editors:
|
||||
# WinEdt
|
||||
*.bak
|
||||
*.sav
|
||||
|
||||
# Texpad
|
||||
.texpadtmp
|
||||
|
||||
# LyX
|
||||
*.lyx~
|
||||
|
||||
# Kile
|
||||
*.backup
|
||||
|
||||
# gummi
|
||||
.*.swp
|
||||
|
||||
# KBibTeX
|
||||
*~[0-9]*
|
||||
|
||||
# TeXnicCenter
|
||||
*.tps
|
||||
|
||||
# auto folder when using emacs and auctex
|
||||
./auto/*
|
||||
*.el
|
||||
|
||||
# expex forward references with \gathertags
|
||||
*-tags.tex
|
||||
|
||||
# standalone packages
|
||||
*.sta
|
||||
|
||||
# Makeindex log files
|
||||
*.lpz
|
||||
|
||||
# xwatermark package
|
||||
*.xwm
|
||||
|
||||
# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
|
||||
# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
|
||||
# Uncomment the next line to have this generated file ignored.
|
||||
#*Notes.bib
|
||||
|
||||
# Draw.io backup files
|
||||
*.bkp
|
||||
*.dtmp
|
||||
|
||||
Binary file not shown.
@@ -0,0 +1,33 @@
|
||||
\documentclass[12pt,letterpaper]{report}
|
||||
\usepackage{init}
|
||||
\usepackage{import}
|
||||
|
||||
\newcommand{\E}{\mathbb{E}}
|
||||
\newcommand{\K}{\mathbb{K}}
|
||||
\newcommand{\dist}{\text{dist}}
|
||||
\newcommand{\lcm}{\text{lcm}}
|
||||
\newcommand{\characteristic}{\text{char }}
|
||||
\newcommand{\gf}{\text{GF}}
|
||||
\newcommand{\fix}{\text{fix}}
|
||||
\newcommand{\gal}{\text{Gal}}
|
||||
\setcounter{chapter}{-1}
|
||||
\author{Alexander J. Clarke}
|
||||
\title{Abstract Algebra Theorems and Definitions}
|
||||
\begin{document}
|
||||
\maketitle
|
||||
\clearpage
|
||||
\begin{center}
|
||||
\thispagestyle{empty}
|
||||
\vspace*{\fill}
|
||||
All theorems, corollaries, lemmas, remarks, and asides are direct quotes from Contemporary Abstract Algebra, 8th Edition, by Joseph A. Gallian
|
||||
\vspace*{\fill}
|
||||
\end{center}
|
||||
\tableofcontents
|
||||
|
||||
\import{part-1/}{part-1.tex}
|
||||
\import{part-2/}{part-2.tex}
|
||||
\import{part-3/}{part-3.tex}
|
||||
\import{part-4/}{part-4.tex}
|
||||
\import{part-5/}{part-5.tex}
|
||||
|
||||
\end{document}
|
||||
@@ -0,0 +1,18 @@
|
||||
# Abstract Algebra Theorems and Definitions
|
||||
This repository houses the LaTeX code (and the corresponding generated [PDF](Abstract-Algebra-Theorems-and-Definitions.pdf))
|
||||
that generates a packet of all of the theorems and definitions contained within
|
||||
[Contemporary Abstract Algebra, 8th Edition, by Joseph A. Gallian](https://a.co/d/eLi1WCJ).
|
||||
|
||||
## Purpose
|
||||
The purpose of this packet is to provide a useful "cheat sheet" for viewing all of the theorems and definitions within the
|
||||
textbook for easy reference when writing proofs.
|
||||
|
||||
## What's Included?
|
||||
This packet includes all of the following from the textbook:
|
||||
|
||||
* Definitions
|
||||
* Theorems
|
||||
* Corollaries
|
||||
* Lemmas
|
||||
* Remarks
|
||||
* Select Examples that contain a definition or theorem
|
||||
@@ -0,0 +1,86 @@
|
||||
\ProvidesPackage{init}
|
||||
|
||||
\usepackage{import}
|
||||
\usepackage[utf8]{inputenc}
|
||||
\usepackage{pgfplots}
|
||||
\usepackage[english]{babel}
|
||||
\usepackage{amsthm}
|
||||
\usepackage{thmtools}
|
||||
\usepackage{hyperref}
|
||||
\usepackage{cancel}
|
||||
\usepackage{mathtools}
|
||||
\usepackage{amsmath}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{amssymb}
|
||||
\usepackage{graphicx}
|
||||
\usepackage{relsize}
|
||||
\usepackage{listings}
|
||||
\graphicspath{ {./images/} }
|
||||
\usepackage{array}
|
||||
\usepackage{tikz}
|
||||
\usetikzlibrary{arrows}
|
||||
\usepackage[left=2cm, right=2.5cm, top=2.5cm, bottom=2.5cm]{geometry}
|
||||
\usepackage{enumitem}
|
||||
\usepackage{mathrsfs}
|
||||
|
||||
% Math Functions
|
||||
\newcommand{\limx}[2]{\displaystyle\lim\limits_{#1 \to #2}}
|
||||
\newcommand{\st}{\ \text{s.t.}\ }
|
||||
\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}
|
||||
\newcommand{\dotp}{\dot{\mathcal{P}}}
|
||||
\newcommand{\dotq}{\dot{\mathcal{Q}}}
|
||||
\newcommand{\Int}[1]{\text{int}\left(#1\right)}
|
||||
\newcommand{\cl}[1]{\text{cl}\left(#1\right)}
|
||||
\newcommand{\bd}[1]{\text{bd}\left(#1\right)}
|
||||
\newcommand{\lr}[1]{\langle #1 \rangle)}
|
||||
\newcommand{\lspan}[1]{\text{span}\left(#1\right)}
|
||||
\newcommand{\ldim}[1]{\text{dim}\left(#1\right)}
|
||||
\newcommand{\nullity}[1]{\text{nullity}\left(#1\right)}
|
||||
\newcommand{\rank}[1]{\text{rank}\left(#1\right)}
|
||||
\newcommand{\ldet}[1]{\text{det}\left(#1\right)}
|
||||
\newcommand{\ltr}[1]{\text{tr}\left(#1\right)}
|
||||
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
|
||||
\DeclareMathOperator{\sign}{sgn}
|
||||
\renewcommand{\qedsymbol}{$\blacksquare$}
|
||||
|
||||
% Special Sets
|
||||
\newcommand{\R}{\mathbb{R}}
|
||||
\newcommand{\N}{\mathbb{N}}
|
||||
\newcommand{\Q}{\mathbb{Q}}
|
||||
\newcommand{\C}{\mathbb{C}}
|
||||
\newcommand{\Z}{\mathbb{Z}}
|
||||
\newcommand{\F}{\mathbb{F}}
|
||||
|
||||
% Theorem Styles
|
||||
\newtheoremstyle{break}% name
|
||||
{}% Space above, empty = `usual value'
|
||||
{}% Space below
|
||||
{}% Body font
|
||||
{}% Indent amount (empty = no indent, \parindent = para indent)
|
||||
{\bfseries}% Thm head font
|
||||
{}% Punctuation after thm head
|
||||
{\newline}% Space after thm head: \newline = linebreak
|
||||
{}% Thm head spec
|
||||
\makeatletter
|
||||
\def\thmhead@plain#1#2#3{%
|
||||
\thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2} }%
|
||||
\thmnote{{\the\thm@notefont\bfseries\itshape#3}}}
|
||||
\let\thmhead\thmhead@plain
|
||||
\makeatother
|
||||
\newtheoremstyle{case}{}{}{}{}{}{:}{ }{}
|
||||
\theoremstyle{case}
|
||||
\newtheorem{case}{Case}
|
||||
\theoremstyle{break}
|
||||
\newtheorem{definition}{Definition}[section]
|
||||
\newtheorem{theorem}{Theorem}[section]
|
||||
\newtheorem{corollary}{Corollary}[section]
|
||||
\newtheorem{lemma}[theorem]{Lemma}
|
||||
\newtheorem*{lem}{Lemma}
|
||||
\newtheorem*{remark}{Remark }
|
||||
\newtheorem*{aside}{}
|
||||
\newtheorem*{example}{Example}
|
||||
|
||||
% Formatting
|
||||
\setlist[enumerate]{font=\bfseries}
|
||||
|
||||
|
||||
@@ -0,0 +1,7 @@
|
||||
\chapter{Preliminaries}
|
||||
\subimport{./}{properties-of-integers.tex}
|
||||
\subimport{./}{modular-arithmetic.tex}
|
||||
\subimport{./}{complex-numbers.tex}
|
||||
\subimport{./}{mathematical-induction.tex}
|
||||
\subimport{./}{equivalence-relations.tex}
|
||||
\subimport{./}{functions-mappings.tex}
|
||||
@@ -0,0 +1,14 @@
|
||||
\section{Complex Numbers}
|
||||
\begin{theorem}[Properties of Complex Numbers]
|
||||
\hfill
|
||||
\begin{enumerate}
|
||||
\item Closure under addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
|
||||
\item Closure under multiplication: $(a + bi)(c + di) = (ac) + (ad)i + (bc)i + (bd)i^2 = (ac - bd) + (ad + bc)i$
|
||||
\item Closure under division ($c + di \neq 0$): $\displaystyle\frac{(a + bi)}{(c + di)} = \frac{(a + bi)}{(c + di)}\frac{(c - di)}{(c - di)}=\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} = \frac{(ac + bd)}{c^2 + d^2} + \frac{(bc - ad)}{c^2 + d^2}i$
|
||||
\item Complex conjugation: $(a + bi)(a - bi) = a^2 + b^2$
|
||||
\item Inverses: For every nonzero complex number $a + bi$ there is a complex number $c + di$ such that $(a + bi)(c + di)=1$. (That is, $(a + bi)^{-1}$ exists in $\C$.)
|
||||
\item Powers: For every complex number $a + bi = r(\cos\theta + i \sin \theta)$ and every positive integer $n$, we have $(a + bi)^n = [r(\cos \theta + i \sin \theta)]^n = r^n(\cos n \theta + i \sin n \theta)$.
|
||||
\item Radicals: For every complex number $a + bi = r(\cos \theta + i \sin \theta)$ and every positive integer $n$, we have $\displaystyle(a + bi)^{\frac{1}{n}} = [r(\cos \theta + i \sin \theta)]^{\frac{1}{n}} = r^{\frac{1}{n}}(\cos\frac{\theta}{n} + i \sin \frac{\theta}{n})$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
@@ -0,0 +1,19 @@
|
||||
\section{Equivalence Relations}
|
||||
|
||||
\begin{definition}[Equivalence Relation]
|
||||
An \textit{equivalence relation} on a set $S$ is a set $R$ of ordered pairs of elements of $S$ such that
|
||||
|
||||
\begin{enumerate}
|
||||
\item $(a, a) \in R$ for all $a \in S$ (reflexive property).
|
||||
\item $(a, b) \in R$ implies $(b, a) \in R$ (symmetric property).
|
||||
\item $(a, b) \in R$ and $(b, c) \in R$ imply $(a, c) \in R$ (transitive property).
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Partition]
|
||||
A \textit{partition} of a set $S$ is a collection of nonempty disjoint subsets of $S$ whose union is $S$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Equivalence Classes Partition]
|
||||
The equivalence classes of an equivalence relation on a set $S$ constitute a partition of $S$. Conversely, for any partition $P$ of $S$, there is an equivalence relation on $S$ whose equivalence classes are the elements of $P$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,27 @@
|
||||
\section{Functions (Mappings)}
|
||||
|
||||
\begin{definition}[Function (Mapping)]
|
||||
A \textit{function} (or \textit{mapping}) $\phi$ from a set $A$ to a set $B$ is a rule that assigns to each element $a$ of $A$ exactly one element $b$ of $B$. The set $A$ is called the \textit{domain of $\phi$}, and $B$ is called the \textit{range of $\phi$}. If $\phi$ assigns $b$ to $a$, then $b$ is called the \textit{image of $a$ under $\phi$}. The subset of $B$ comprising all the images of elements of $A$ is called the \textit{image of $A$ under $\phi$}.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Composition of Functions]
|
||||
Let $\phi: A \to B$ and $\psi: B \to C$. The \textit{composition $\psi\phi$} is the mapping from $A$ to $C$ defined by $(\psi\phi)(a) = \psi(\phi(a))$ for all $a$ in $A$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[One-to-One Function]
|
||||
A function $\phi$ from a set $A$ is called \textit{one-to-one} if for every $a_1, a_2 \in A,\ \phi(a_1) = \phi(a_2)$ implies $a_1 = a_2$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Functions from A onto B]
|
||||
A function $\phi$ from a set $A$ to a set $B$ is said to be \textit{onto $B$} if each element of $B$ is the image of at least one element of $A$. In symbols, $\phi: A \to B$ is onto if for each $b$ in $B$ there is at least one $a$ in $A$ such that $\phi(a) = b$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Properties of Functions]
|
||||
Given functions $\alpha: A \to B$, $\beta: B \to C$, and $\gamma: C \to D$, then
|
||||
\begin{enumerate}
|
||||
\item $\gamma(\beta\alpha) = (\gamma\beta)\alpha$ (associativity).
|
||||
\item If $\alpha$ and $\beta$ are one-to-one, then $\beta\alpha$ is one-to-one.
|
||||
\item If $\alpha$ and $\beta$ are onto, then $\beta\alpha$ is onto.
|
||||
\item If $\alpha$ is one-to-one and onto, then there is a function $\alpha^{-1}$ from $B$ onto $A$ such that $(\alpha^{-1}\alpha)(a) = a$ for all $a$ in $A$ and $(\alpha\alpha^{-1})(b) = b$ for all $b$ in $B$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{Mathematical Induction}
|
||||
|
||||
\begin{theorem}[First Principle of Mathematical Induction]
|
||||
Let $S$ be a set of integers containing $a$. Suppose $S$ has the property that whenever some integer $n \geq a$ belongs to $S$, then the integer $n + 1$ also belongs to $S$. Then, $S$ contains every integer greater than or equal to $a$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Second Principle of Mathematical Induction]
|
||||
Let $S$ be a set of integers containing $a$. Suppose $S$ has the property that $n$ belongs to $S$ whenever every integer less than $n$ and greater than or equal to $a$ belongs to $S$. Then, $S$ contains every integer greater than or equal to $a$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1 @@
|
||||
\section{Modular Arithmetic}
|
||||
@@ -0,0 +1,33 @@
|
||||
\section{Properties of Integers}
|
||||
\begin{aside}[Well Ordering Principle]
|
||||
Every nonempty set of positive integers contains a smallest number.
|
||||
\end{aside}
|
||||
|
||||
\begin{theorem}[Division Algorithm]
|
||||
Let $a$ and $b$ be integers with $b > 0$. then there exist unique integers $q$ and $r$ with the property that $a = bq + r$, where $0 \leq r < b$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Greatest Common Divisor, Relatively Prime Integers]
|
||||
The \textit{greatest common divisor} of two nonzero integers $a$ and $b$ is the largest of all common divisors of $a$ and $b$. We denote this integer by $\gcd(a, b)$. When $\gcd(a, b) = 1$, we say that $a$ and $b$ are \textit{relatively prime}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[GCD Is a Linear Combination]
|
||||
for any nonzero integers $a$ and $b$, there exist integers $s$ and $t$ such that $\gcd(a, b)=as+bt$. Moreover, $\gcd(a,b)$ is the smallest positive integer of the form $as + bt$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}
|
||||
If $a$ and $b$ are relatively prime, then there exist integers $s$ and $t$ such that $as + bt = 1$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{lemma}[Euclid's Lemma \text{\normalfont $p\ \vert\ ab$ implies $p\ \vert\ a$ or $p\ \vert\ b$}]
|
||||
If $p$ is a prime that divides $ab$, then $p$ divides $a$ or $p$ divides $b$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[Fundamental Theorem of Arithmetic]
|
||||
Every integer greater than 1 is a prime or a product of primes. this product is unique, except for the order in which the factors appear. That is, if $n = p_1p_2\dots p_r$ and $n=q_1q_2\dots q_s$, where the $p$'s and $q$'s are primes, then $r = s$ and, after renumbering the $q$'s, we have $p_i = q_i$ for all $i$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Least Common Multiple]
|
||||
The \textit{least common multiple} of two nonzero integers $a$ and $b$ is the smallest positive integer that is a multiple of both $a$ and $b$. We will denote this integer by $\lcm(a, b)$.
|
||||
\end{definition}
|
||||
|
||||
@@ -0,0 +1,3 @@
|
||||
\part{Integers and Equivalence Relations}
|
||||
\subimport{chapters/chapter-0}{chapter-0.tex}
|
||||
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Group Homomorphisms}
|
||||
\subimport{./}{definition-and-examples.tex}
|
||||
\subimport{./}{properties-of-homomorphisms.tex}
|
||||
\subimport{./}{the-first-isomorphism-theorem.tex}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{Definition and Examples}
|
||||
|
||||
\begin{definition}[Group Homomorphism]
|
||||
A \textit{homomorphism} $\phi$ from a group $G$ to a group $\overline{G}$ is a mapping from $G$ into $\overline{G}$ that preserves the group operation; that is, $\phi(ab) = \phi(a)\phi(b)$ for all $a, b$ in $G$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Kernel of a Homomorphism]
|
||||
The \textit{kernel} of a homomorphism $\phi$ from a group $G$ to a group with identity $e$ is the set $\{x \in G\ \vert\ \phi(x)=e\}$. The kernel of $\phi$ is denoted by $\ker\phi$.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,32 @@
|
||||
\section{Properties of Homomorphisms}
|
||||
|
||||
\begin{theorem}[Properties of Elements Under Homomorphisms]
|
||||
Let $\phi$ be a homomorphism from a group $G$ to a group $\overline{G}$ and let $g$ be an element of $G$. Then
|
||||
\begin{enumerate}
|
||||
\item $\phi$ carries the identity of $G$ to $\overline{G}$.
|
||||
\item $\phi(g^n)=(\phi(g))^n$ for all $n$ in $\Z$.
|
||||
\item If $\abs{g}$ is finite, then $\abs{\phi(g)}$ divides $\abs{g}$.
|
||||
\item $\ker\phi$ is a subgroup of $G$.
|
||||
\item $\phi(a) = \phi(b)$ if and only if $a\ker\phi = b\ker\phi$.
|
||||
\item If $\phi(g) = g'$, then $\phi^{-1}(g') = \{x \in G\ \vert\ \phi(x) = g'\} = g\ker\phi$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Properties of Subgroups Under Homomorphisms]
|
||||
Let $\phi$ be a homomorphism from a group $G$ to a group $\overline{G}$ and let $H$ be a subgroup of $G$. Then
|
||||
\begin{enumerate}
|
||||
\item $\phi(H) = \{\phi(h)\ \vert\ h \in H\}$ is a subgroup of $\overline{G}$.
|
||||
\item If $H$ is cyclic, then $\phi(H)$ is cyclic.
|
||||
\item If $H$ is Abelian, then $\phi(H)$ is Abelian.
|
||||
\item If $H$ is normal in $G$, then $\phi(H)$ is normal in $\phi(G)$.
|
||||
\item If $\abs{\ker\phi} = n$, then $\phi$ is an $n$-to-1 mapping from $G$ onto $\phi(G)$.
|
||||
\item If $\abs{H} = n$, then $\abs{\phi(H)}$ divides $n$.
|
||||
\item If $\overline{K}$ is a subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K})=\{k \in G\ \vert\ \phi(k) \in \overline{K}\}$ is a subgroup of $G$.
|
||||
\item If $\overline{K}$ is a normal subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K})=\{ k \in G\ \vert\ \phi(k) \in \overline{K}\}$ is a normal subgroup of $G$.
|
||||
\item If $\phi$ is onto and $\ker\phi = \{e\}$, then $\phi$ is an isomorphism from $G$ to $\overline{G}$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Kernels Are Normal]
|
||||
Let $\phi$ be a group homomorphism from $G$ to $\overline{G}$. Then $\ker\phi$ is a normal subgroup of $G$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{The First Isomorphism Theorem}
|
||||
|
||||
\begin{theorem}[First Isomorphism Theorem (Jordan, 1870)]
|
||||
Let $\phi$ be a group homomorphism from $G$ to $\overline{G}$. Then the mapping from $G/\ker\phi$ to $\phi(G)$, given by $g\ker\phi \to \phi(g)$, is an isomorphism. In symbols, $G/\ker\phi \approx \phi(G)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}
|
||||
If $\phi$ is a homomorphism from a finite group $G$ to $\overline{G}$, then $\abs{\phi(G)}$ divides $\abs{G}$ and $\abs{\overline{G}}$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[Normal Subgroups Are Kernels]
|
||||
Every normal subgroup of a group $G$ is the kernel of a homomorphism of $G$. In particular, a normal subgroup $N$ is the kernel of the mapping $g \to gN$ from $G$ to $G/N$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Fundamental Theorem of Finite Abelian Groups}
|
||||
\subimport{./}{the-fundamental-theorem.tex}
|
||||
\subimport{./}{the-isomorphism-classes-of-abelian-groups.tex}
|
||||
\subimport{./}{proof-of-the-fundamental-theorem.tex}
|
||||
@@ -0,0 +1,17 @@
|
||||
\section{Proof of the Fundamental Theorem}
|
||||
|
||||
\begin{lemma}
|
||||
Let $G$ be a finite Abelian group of order $p^nm$, where $p$ is a prime that does not divide $m$. Then $G = H \times K$, where $H = \{x \in G\ \vert\ x^{p^n} =e\}$ and $K =\{x \in G\ \vert\ x^m = e\}$. Moreover, $\abs{H}=p^n$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{lemma}
|
||||
Let $G$ be an Abelian group of prime-power order and let $a$ be an element of maximum order in $G$. Then $G$ can be written in the form $\lr{a} \times K$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{lemma}
|
||||
A finite Abelian group of prime-power order is an internal direct product of cyclic groups.
|
||||
\end{lemma}
|
||||
|
||||
\begin{lemma}
|
||||
Suppose that $G$ is a finite Abelian group of prime-power order. If $G=H_1 \times H_2 \times \dots \times H_m$ and $G=K_1 \times K_2 \times \dots \times K_n$, where the $H$'s and $K$'s are nontrivial cyclic subgroups with $\abs{H_1} \geq \abs{H_2} \geq \dots \geq \abs{H_m}$ and $\abs{K_1} \geq \abs{K_2} \geq \dots \geq \abs{K_n}$, then $m=n$ and $\abs{H_i} = \abs{K_i}$ for all $i$.
|
||||
\end{lemma}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{The Fundamental Theorem}
|
||||
|
||||
\begin{theorem}[Fundamental Theorem of Finite Abelian Groups]
|
||||
Every finite Abelian group is a direct product of cyclic groups of prime-power order. Moreover, the number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,16 @@
|
||||
\section{The Isomorphism Classes of Abelian Groups}
|
||||
|
||||
\begin{remark}[Greedy Algorithm for an Abelian Group of Order $\mathbf{p^n}$]
|
||||
The Fundamental Theorem is extremely powerful. As an application, we can use it as an algorithm for constructing all Abelian groups of any order. Let's look at Abelian groups of a certain order $n$, where $n$ has two or more distinct prime divisors.
|
||||
\begin{enumerate}
|
||||
\item Compute the orders of the elements of the group $G$
|
||||
\item Select an element $a_1$ of maximum order and define $G_1 = \lr{a_1}$. Set $i = 1$.
|
||||
\item If $\abs{G} = \abs{G_i}$, stop. Otherwise, replace $i$ by $i + 1$.
|
||||
\item Select an element $a_i$ of maximum order $p^k$ such that $p^k \leq \abs{G}/\abs{G_{i-1}}$ and none of $a_i, a^p_i,a^{p^2}_i, \dots, a^{p^{k-1}}_i$ is in $G_{i-1}$, and define $G_i=G_{i-1} \times \lr{a_i}$.
|
||||
\item Return to step 3.
|
||||
\end{enumerate}
|
||||
\end{remark}
|
||||
|
||||
\begin{corollary}[Existence of Subgroups of Abelian Groups]
|
||||
If $m$ divides the order of a finite Abelian group $G$, then $G$ has a subgroup of order $m$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Groups}
|
||||
\subimport{./}{definition-and-examples-of-groups.tex}
|
||||
\subimport{./}{elementary-properties-of-groups.tex}
|
||||
@@ -0,0 +1,14 @@
|
||||
\section{Definition and Examples of Groups}
|
||||
|
||||
\begin{definition}[Binary Operation]
|
||||
Let $G$ be a set. A \textit{binary operation} on $G$ is a function that assigns each ordered pair of elements of $G$ an element of $G$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Group]
|
||||
Let $G$ be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair $(a, b)$ of elements of $G$ an element in $G$ denoted by $ab$. We say $G$ is a \textit{group} under this operation if the following three properties are satisfied.
|
||||
\begin{enumerate}
|
||||
\item \textit{Associativity}. The operation is associative; that is, $(ab)c = a(bc)$ for all $a,b,c$ in $G$.
|
||||
\item \textit{Identity}. There is an element $e$ (called the \textit{identity}) in $G$ such that $ae = ea = a$ for all $a$ in $G$.
|
||||
\item \textit{Inverses}. For each element $a$ in $G$, there is an element $b$ in $G$ (called an \textit{inverse} of $a$) such that $ab = ba = e$.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
@@ -0,0 +1,17 @@
|
||||
\section{Elementary Properties of Groups}
|
||||
|
||||
\begin{theorem}[Uniqueness of the Identity]
|
||||
In a group $G$, there is only one identity element.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Cancellation]
|
||||
In a group $G$, the right and left cancellation laws hold; that is, $ba = ca$ implies $b = c$, and $ab = ac$ implies $b = c$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Uniqueness of Inverses]
|
||||
For each element $a$ in a group $G$, there is a unique element $b$ in $G$ such that $ab = ba = e$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Socks-Shoes Property]
|
||||
For group elements $a$ and $b$, $(ab)^{-1} = b^{-1}a^{-1}$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Finite Groups; Subgroups}
|
||||
\subimport{./}{subgroup-tests.tex}
|
||||
\subimport{./}{terminology-and-notation.tex}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Terminology and Notation}
|
||||
|
||||
\begin{definition}[Order of a Group]
|
||||
The number of elements of a group (finite or infinite) is called its \textit{order}. We will use $\abs{G}$ to denote the order of $G$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Order of an Element]
|
||||
The \textit{order} of an element $g$ in a group $G$ is the smallest positive integer $n$ such that $g^n = e$. (In additive notation, this would be $ng = 0$.) If no such integer exists, we say that $g$ has \textit{infinite order}. The order of an element $g$ is denoted by $\abs{g}$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Subgroup]
|
||||
If a subset $H$ of a group $G$ is itself a group under the operation of $G$, we say that $H$ is a \textit{subgroup} of $G$.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,36 @@
|
||||
\section{Subgroup Tests}
|
||||
|
||||
\begin{theorem}[One-Step Subgroup Test]
|
||||
Let $G$ be a group and $H$ a nonempty subset of $G$. If $ab^{-1}$ is in $H$ whenever $a$ and $b$ are in $H$, then $H$ is a subgroup of $G$. (In additive notation, if $a - b$ is in $H$ whenever $a$ and $b$ are in $H$, then $H$ is a subgroup of $G$.)
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Two-Step Subgroup Test]
|
||||
Let $G$ be a group and let $H$ be a nonempty subset of $G$. If $ab$ is in $H$ whenever $a$ and $b$ are in $H$ ($H$ is closed under the operation), and $a^{-1}$ is in $H$ whenever $a$ is in $H$ ($H$ is closed under taking inverses), then $H$ is a subgroup of $G$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Finite Subgroup Test]
|
||||
Let $H$ be a nonempty finite subset of a group $G$. If $H$ is closed under the operation of $G$, then $H$ is a subgroup of $G$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[$\mathbf{\lr{a}}$ Is a Subgroup]
|
||||
Let $G$ be a group, and let $a$ be any element of $G$. Then, $\lr{a}$ is a subgroup of $G$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Center of a Group]
|
||||
The \textit{center}, $Z(G)$, of a group $G$ is the subset of elements in $G$ that commute with every element of $G$. In symbols,
|
||||
\[ Z(G) = \{a \in G\ \vert\ ax = xa,\ \forall\ x \in G\} \]
|
||||
[The notation $Z(G)$ comes from the fact that the German word for center is \textit{Zentrum}. The term was coined by J.A. de Séguier in 1904.]
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Center Is a Subgroup]
|
||||
The center of a group $G$ is a subgroup of $G$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Centralizer of $\mathbf{a}$ in $\mathbf{G}$]
|
||||
Let $a$ be a fixed element of a group $G$. The \textit{centralizer of $a$ in $G$}, $C(a)$, is the set of all elements in $G$ that commute with $a$. In symbols,
|
||||
\[ C(a) = \{g \in G\ \vert\ ga = ag\} \]
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[$\mathbf{C(a)}$ Is a Subgroup]
|
||||
For each $a$ in a group $G$, the centralizer of $a$ is a subgroup of $G$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Cyclic Groups}
|
||||
\subimport{./}{properties-of-cyclic-groups.tex}
|
||||
\subimport{./}{classification-of-subgroups-of-cyclic-groups.tex}
|
||||
@@ -0,0 +1,17 @@
|
||||
\section{Classification of Subgroups of Cyclic Groups}
|
||||
|
||||
\begin{theorem}[Fundamental Theorem of Cyclic Groups]
|
||||
Every subgroup of a cyclic group is cyclic. Moreover, if $\abs{\lr{a}} = n$, then the order of any subgroup of $\lr{a}$ is a divisor of $n$; and, for each, positive divisor $k$ of $n$, the group $\lr{a}$ has exactly one subgroup of order $k$ -- namely, $\lr{a^{n/k}}$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Subgroups of $\mathbf{\Z_n}$]
|
||||
For each positive divisor $k$ of $n$, the set $\lr{n/k}$ is the unique subgroup of $\Z_n$ of order $k$; moreover, these are the only subgroups of $\Z_n$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[Number of Elements of Each Order in a Cyclic Group]
|
||||
If $d$ is a positive divisor of $n$, the number of elements of order $d$ in a cyclic group of order $n$ is $\phi(d)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Number of Elements of Order $\mathbf{d}$ in a Finite Group]
|
||||
In a finite group, the number of elements of order $d$ is a multiple of $\phi(d)$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,33 @@
|
||||
\section{Properties of Cyclic Groups}
|
||||
|
||||
\begin{theorem}[Criterion for $\mathbf{a^i=a^j}$]
|
||||
Let $G$ be a group, and let $a$ belong to $G$. If $a$ has infinite order, then $a^i = a^j$ if and only if $i = j$. If $a$ has finite order, say, $n$, then $\lr{a} = \{e, a, a^2, \dots, a^{n - 1}\}$ and $a^i = a^j$ if and only if $n$ divides $i - j$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[$\mathbf{\abs{a}=\abs{\lr{a}}}$]
|
||||
For any group element $a$, $\abs{a} = \abs{\lr{a}}$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[$\mathbf{a^k = e}$ Implies That $\mathbf{\abs{a}}$ Divides $\mathbf{k}$]
|
||||
Let $G$ be a group and let $a$ be an element of order $n$ in $G$. If $a^k = e$, then $n$ divides $k$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[$\mathbf{\lr{a^k} = \lr{a^{\textbf{gcd}(n,k)}}}$ and $\mathbf{\abs{a^k} = n/\textbf{gcd}(n,k)}$]
|
||||
Let $a$ be an element of order $n$ in a gruop and let $k$ be a positive integer. Then $\lr{a^k} = \lr{a^{\gcd(n,k)}}$ and $\abs{a^k} = n / \gcd(n,k)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Orders of Elements in Finite Cyclic Groups]
|
||||
In a finite cyclic group, the order of an element divides the order of the group.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Criterion for $\mathbf{\lr{a^i} = \lr{a^j}}$ and $\mathbf{\abs{a^i} = \abs{a^j}}$]
|
||||
Let $\abs{a} = n$. Then $\lr{a^i} = \lr{a^j}$ if and only if $\gcd(n, i) = \gcd(n,j)$, and $\abs{a^i} = \abs{a^j}$ if and only if $\gcd(n,i) = \gcd(n,j)$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Generators of Finite Cyclic Groups]
|
||||
Let $\abs{a} = n$. Then $\lr{a} = \lr{a^j}$ if and only if $\gcd(n,j) = 1$, and $\abs{a} = \abs{\lr{a^j}}$ if and only if $\gcd(n,j) = 1$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Generators of $\mathbf{\Z_n}$]
|
||||
An integer $k$ in $\Z_n$ is a generator of $Z_n$ if and only if $\gcd(n,k) = 1$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Permutation Groups}
|
||||
\subimport{./}{definition-and-notation.tex}
|
||||
\subimport{./}{cycle-notation.tex}
|
||||
\subimport{./}{properties-of-permutations.tex}
|
||||
@@ -0,0 +1,28 @@
|
||||
\section{Cycle Notation}
|
||||
\begin{definition}
|
||||
Consider the permutation
|
||||
\[ \alpha = \begin{bmatrix}
|
||||
1 & 2 & 3 & 4 & 5 & 6 \\
|
||||
2 & 1 & 4 & 6 & 5 & 3
|
||||
\end{bmatrix}\]
|
||||
|
||||
The assignment of values is as follows:
|
||||
|
||||
\begin{align*}
|
||||
1 & \mapsto 2 \\
|
||||
2 & \mapsto 1 \\
|
||||
3 & \mapsto 4 \\
|
||||
4 & \mapsto 6 \\
|
||||
5 & \mapsto 5 \\
|
||||
6 & \mapsto 3
|
||||
\end{align*}
|
||||
|
||||
Although mathematically satisfactory, such diagrams are cumbersome. Instead, we leave out the arrows and simply write $\alpha = (1,2)(3,4,6)(5)$.
|
||||
|
||||
It is also worth noting that an expression of the form $(a_1, a_2, \dots, a_m)$ is called a \textit{cycle of length $m$}, or an \textit{$m$-cycle}.
|
||||
\end{definition}
|
||||
\begin{example}
|
||||
To multiply cycles, consider the following permutations from $S_8$. Let $\alpha = (13)(27)(456)(8)$ and $\beta = (1237)(648)(5)$. (When the domain consists of single-digit integers, it is common practice to omit the commas between the digits.) What is the cycle form of $\alpha\beta$? Of course, one could say that $\alpha\beta = (13)(27)(456)(8)(1237)(648)(5)$, but it is usually more desirable to express a permutation in a \textit{disjoint} cycle form (that is, the various cycles have no number in common). Well, keeping in mind that function composition is done from right to left and that each cycle that does not contain a symbol fixes the symbol, we observe that $(5)$ fixes 1; $(648)$ fixes $1$; $(1237)$ sends 1 to 2, $(8)$ fixes 2; $(456)$ fixes 2; $(27)$ sends 2 to 7; and $(13)$ fixes 7. So the net effect of $\alpha\beta$ is to send 1 to 7. Thus, we begin $\alpha\beta=(17\dots)\dots$. Now, repeating the entire process beginning with 7, we have, cycle by cycle, right to left,
|
||||
\[ 7 \to 7 \to 7 \to 1 \to 1 \to 1 \to 1 \to 3, \]
|
||||
so that $\alpha\beta = (173\dots)\dots$. Ultimately, we have $\alpha\beta = (1732)(48)(56)$. The import thing to bear in mind when multiplying cycles is to "keep moving" from one cycle to the next from right to left.
|
||||
\end{example}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Definition and Notation}
|
||||
|
||||
\begin{definition}[Permutation of $\mathbf{A}$, Permutation Group of $\mathbf{A}$]
|
||||
A \textit{permutation} of a set $A$ is a function from $A$ to $A$ that is both one-to-one and onto. A \textit{permutation group} of a set $A$ is a set of permutations of $A$ that forms a group under function composition.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,43 @@
|
||||
\section{Properties of Permutations}
|
||||
|
||||
\begin{theorem}[Products of Disjoint Cycles]
|
||||
Every permutation of a finite set can be written as a cycle or as a product of disjoint cycles.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Disjoint Cycles Commute]
|
||||
If the pair of cycles $\alpha = (a_1, a_2, \dots, a_m)$ and $\beta = (b_1, b_2, \dots, b_n)$ have no entries in common, then $\alpha\beta = \beta\alpha$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Order of a Permutation (Ruffini, 1799)]
|
||||
The order of a permutation of a finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Product of 2-Cycles]
|
||||
Every permutation in $S_n,\ n>1$ is a product of 2-cycles.
|
||||
\end{theorem}
|
||||
|
||||
\begin{lem}
|
||||
If $\varepsilon = \beta_1\beta_2\dots\beta_r$, where the $\beta$'s are 2-cycles, then $r$ is even.
|
||||
\end{lem}
|
||||
|
||||
\begin{theorem}[Always Even or Always Odd]
|
||||
If a permutation $\alpha$ can be expressed as a product of an even (odd) number of 2-cycles, then every decomposition of $\alpha$ into a product of 2-cycles must have an even (odd) number of 2-cycles. In symbols, if
|
||||
\[ \alpha = \beta_1\beta_2\dots\beta_r\ \ \ \ \text{and}\ \ \ \ \alpha=\gamma_1\gamma_2\dots\gamma_s, \]
|
||||
where the $\beta$'s and the $\gamma$'s are 2-cycles, then $r$ and $s$ are both even or both odd.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Even and Odd Permutations]
|
||||
A permutation that can be expressed as a product of an even number of 2-cycles is called an \textit{even} permutation. A permutation that can be expressed as a product of an odd number of 2-cycles is called an \textit{odd} permutation.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Even Permutations Form a Group]
|
||||
The set of even permutations in $S_n$ forms a subgroup of $S_n$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Alternating Group of Degree $\mathbf{n}$]
|
||||
The group of even permutations of $n$ symbols is denoted by $A_n$ and is called the \textit{alternating group of degree $n$}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
For $n > 1$, $A_n$ has order $n!/2$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,19 @@
|
||||
\section{Automorphisms}
|
||||
|
||||
\begin{definition}[Automorphism]
|
||||
An isomorphism from a group $G$ onto itself is called an \textit{automorphisms} of $G$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Inner Automorphism Induced by $\mathbf{a}$]
|
||||
Let $G$ be a group, and let $a \in G$. The function $\phi_a$ defined by $\phi_a(x) = axa^{-1}$ for all $x$ in $G$ is called the \textit{inner automorphism of $G$ induced by $a$}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Aut($G$) and Inn($G$) Are Groups]
|
||||
The set of automorphisms of a group and the set of inner automorphisms of a group are both groups under the operation of function composition.
|
||||
|
||||
When $G$ is a group, we use Aut($G$) to denote the set of all automorphisms of $G$ and Inn($G$) to denote the set of all inner automorphisms of $G$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Aut$\mathbf{(\Z_n) \approx U(n)}$]
|
||||
For every positive integer $n$, Aut($\Z_n$) is isomorphic to $U(n)$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Cayley's Theorem}
|
||||
|
||||
\begin{theorem}[Cayley's Theorem (1854)]
|
||||
Every group is isomorphic to a group of permutations.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,5 @@
|
||||
\chapter{Isomorphisms}
|
||||
\subimport{./}{definition-and-examples.tex}
|
||||
\subimport{./}{cayleys-theorem.tex}
|
||||
\subimport{./}{properties-of-isomorphisms.tex}
|
||||
\subimport{./}{automorphisms.tex}
|
||||
@@ -0,0 +1,7 @@
|
||||
\section{Definition and Examples}
|
||||
|
||||
\begin{definition}[Group Isomorphism]
|
||||
An \textit{isomorphism} $\phi$ from a group $G$ to a group $\overline{G}$ is a one-to-one mapping (or function) from $G$ onto $\overline{G}$ that preserves the group operation. That is,
|
||||
\[ \phi(ab) = \phi(a)\phi(b),\ \forall a,b \in G \]
|
||||
If there is an isomorphism from $G$ onto $\overline{G}$, we say that $G$ and $\overline{G}$ are \textit{isomorphic} and write $G \approx \overline{G}$.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,26 @@
|
||||
\section{Properties of Isomorphisms}
|
||||
|
||||
\begin{theorem}[Properties of Isomorphisms Acting on Elements]
|
||||
Suppose that $\phi$ is an isomorphism from a group $G$ onto a group $\overline{G}$. Then
|
||||
\begin{enumerate}
|
||||
\item $\phi$ carries the identity of $G$ to the identity of $\overline{G}$.
|
||||
\item For every integer $n$ and for every group element $a$ in $G$, $\phi(a^n)=[\phi(a)]^n$.
|
||||
\item For any elements $a$ and $b$ in $G$, $a$ and $b$ commute if and only if $\phi(a)$ and $\phi(b)$ commute.
|
||||
\item $G = \lr{a}$ if and only if $\overline{G} = \lr{\phi(a)}$.
|
||||
\item $\abs{a}=\abs{\phi(a)}$ for all $a$ in $G$ (isomorphisms preserve orders).
|
||||
\item For a fixed integer $k$ and a fixed group element $b$ in $G$, the equation $x^k=b$ has the same number of solutions in $G$ as does the equation $x^k = \phi(b)$ in $\overline{G}$.
|
||||
\item If $G$ is finite, then $G$ and $\overline{G}$ have exactly the same number of elements of every order.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Properties of Isomorphisms Acting on Groups]
|
||||
Suppose that $\phi$ is an isomorphism from a group $G$ onto a group $\overline{G}$. Then
|
||||
\begin{enumerate}
|
||||
\item $\phi^{-1}$ is an isomorphisms from $\overline{G}$ onto $G$.
|
||||
\item $G$ is Abelian if and only if $\overline{G}$ is Abelian.
|
||||
\item $G$ is cyclic if and only if $\overline{G}$ is cyclic.
|
||||
\item If $K$ is a subgroup of $G$, then $\phi(K) = \{\phi(k)\ \vert\ k \in K\}$ is a subgroup of $\overline{G}$.
|
||||
\item If $\overline{K}$ is a subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K}) = \{g \in G\ \vert\ \phi(g) \in \overline{K}\}$ is a subgroup of $G$.
|
||||
\item $\phi(Z(G))=Z(\overline{G})$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{An Application of Cosets to Permutation Groups}
|
||||
|
||||
\begin{definition}[Stabilizer of a Point]
|
||||
Let $G$ be a group of permutations of a set $S$. For each $i$ in $S$, let stab$_G(i)=\{\phi \in G\ \vert\ \phi(i) = i\}$. We call stab$_G(i)$ the \textit{stabilizer of $i$ in $G$}.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Orbit of a Point]
|
||||
Let $G$ be a group of permutations of a set $S$. For each $s$ in $S$, let orb$_G(s)=\{\phi(s)\ \vert\ \phi \in G\}$. The set orb$_G(s)$ is a subset of $S$ called the \textit{orbit of $s$ under $G$}. We use $\abs{\text{orb}_G(s)}$ to denote the number of elements in orb$_G(s)$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Orbit-Stabilizer Theorem]
|
||||
Let $G$ be a finite group of permutations of a set $S$. Then, for any $i$ from $S$, $\abs{G} = \abs{\text{orb}_G(i)}\abs{\text{stab}_G(i)}$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,5 @@
|
||||
\chapter{Cosets and Lagrange's Theorem}
|
||||
\subimport{./}{properties-of-cosets.tex}
|
||||
\subimport{./}{lagranges-theorem-and-consequences.tex}
|
||||
\subimport{./}{an-application-of-cosets-to-permutation-groups.tex}
|
||||
\subimport{./}{the-rotation-group-of-a-cube-and-a-soccer-ball.tex}
|
||||
@@ -0,0 +1,37 @@
|
||||
\section{Lagrange's Theorem and Consequences}
|
||||
|
||||
\begin{theorem}[Lagrange's Theorem: $\mathbf{\abs{H} \text{ Divides } \abs{G}}$]
|
||||
If $G$ is a finite group and $H$ is a subgroup of $G$, then $\abs{H}$ divides $\abs{G}$. Moreover, the number of distinct left (right) cosets of $H$ in $G$ is $\abs{G}/\abs{H}$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{remark}
|
||||
A special name and notation have been adopted for the number of left (or right) cosets of a subgroup in a group. The \textit{index} of a subgroup $H$ in $G$ is the number of distinct left cosets of $H$ in $G$. This number is denoted by $\abs{G:H}$.
|
||||
\end{remark}
|
||||
|
||||
\begin{corollary}[$\mathbf{\abs{G:H} = \abs{G}/\abs{H}}$]
|
||||
If $G$ is a finite group and $H$ is a subgroup of $G$, then $\abs{G:H} = \abs{G}/\abs{H}$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[$\mathbf{\abs{a}}$ Divides $\mathbf{\abs{G}}$]
|
||||
In a finite group, the order of each element of the group divides the order of the group.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Groups of Prime Order Are Cyclic]
|
||||
A group of prime order is cyclic.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[$\mathbf{a^{\abs{G}}=e}$]
|
||||
Let $G$ be a finite group, and let $a \in G$. Then, $a^{\abs{G}} = e$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Fermat's Little Theorem]
|
||||
For every integer $a$ and every prime $p$, $a^p \mod p = a \mod p$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[$\mathbf{\abs{HK} = \abs{H}\abs{K}/\abs{H \cap K}}$]
|
||||
For two finite subgroups $H$ and $K$ of a group, define the set $HK = \{hk\ \vert\ h \in H, k \in K\}$. Then $\abs{HK} = \abs{H}\abs{K}/\abs{H \cap K}$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Classification of Groups of order 2$\mathbf{p}$]
|
||||
Let $G$ be a group of order $2p$, where $p$ is a prime greater than 2. Then $G$ is isomorphic to $\Z_{2p}$ or $D_p$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,20 @@
|
||||
\section{Properties of Cosets}
|
||||
|
||||
\begin{definition}[Coset of $\mathbf{H}$ in $\mathbf{G}$]
|
||||
Let $G$ be a group and let $H$ be a nonempty subset of $G$. For any $a \in G$, the set $\{ah\ \vert\ h \in H\}$ is denoted by $aH$. Analogously, $Ha = \{ha\ \vert\ h \in H\}$ and $aHa^{-1} = \{aha^{-1}\ \vert\ h \in H\}$. When $H$ is a subgroup of $G$, the set $aH$ is called the \textit{left coset of $H$ in $G$ containing $a$}, whereas $Ha$ is called the \textit{right coset of $H$ in $G$ containing $a$}. In this case, the element $a$ is called the \textit{coset representative of $aH$ (or $Ha$)}. We use $\abs{aH}$ to denote the number of elements in the set $aH$, and $\abs{Ha}$ to denote the number of elements in $Ha$.
|
||||
\end{definition}
|
||||
|
||||
\begin{lem}[ Properties of Cosets]
|
||||
Let $H$ be a subgroup of $G$, and let $a$ and $b$ belong to $G$. Then,
|
||||
\begin{enumerate}
|
||||
\item $a \in aH$.
|
||||
\item $aH = H$ if and only if $a \in H$.
|
||||
\item $(ab)H = a(bH)$ and $H(ab) = (Ha)b$.
|
||||
\item $aH = bH$ if and only if $a \in bH$.
|
||||
\item $aH = bH$ or $aH \cap bH = \emptyset$.
|
||||
\item $aH = bH$ if and only if $a^{-1}b \in H$.
|
||||
\item $\abs{aH}=\abs{bH}$.
|
||||
\item $aH = Ha$ if and only if $H = aHa^{-1}$.
|
||||
\item $aH$ is a subgroup of $G$ if and only if $a \in H$.
|
||||
\end{enumerate}
|
||||
\end{lem}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{The Rotation Group of a Cube and a Soccer Ball}
|
||||
|
||||
\begin{theorem}[The Rotation Group of a Cube]
|
||||
The group of rotations of a cube is isomorphic to $S_4$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{External Direct Products}
|
||||
\subimport{./}{definition-and-examples.tex}
|
||||
\subimport{./}{properties-of-external-direct-products.tex}
|
||||
\subimport{./}{the-group-of-units-modulo-n-as-an-external-direct-product.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Definition and Examples}
|
||||
|
||||
\begin{definition}[External Direct Product]
|
||||
Let $G_1,G_2,\dots,G_n$ be a finite collection of groups. The \textit{external direct product} of $G_1,G_2,\dots,G_n$, written as $G_1 \oplus G_2 \oplus \dots \oplus G_n$, is the set of all $n$-tuples for which the $i$th component is an element of $G_i$ and the operation is componentwise.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,18 @@
|
||||
\section{Properties of External Direct Products}
|
||||
|
||||
\begin{theorem}[Order of an Element in a Direct Product]
|
||||
The order of an element in a direct product of a finite number of finite groups is the least common multiple of the orders of the component of the element. In symbols,
|
||||
\[ \abs{(g_1,g_2,\dots,g_n)} = \lcm(\abs{g_1},\abs{g_2},\dots,\abs{g_n}) \]
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Criterion for $\mathbf{G \oplus H}$ to be Cyclic]
|
||||
Let $G$ and $H$ be finite cyclic groups. Then $G \oplus H$ is cyclic if and only if $\abs{G}$ and $\abs{H}$ are relatively prime.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Criterion for $\mathbf{G_1 \oplus G_2 \oplus \dots \oplus G_n}$ to Be Cyclic]
|
||||
An external direct product $G_1 \oplus G_2 \oplus \dots \oplus G_n$ of a finite number of finite cyclic groups is cyclic if and only if $\abs{G_i}$ and $\abs{G_j}$ are relatively prime when $i \neq j$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Criterion for $\mathbf{\Z_{n_1n_2\dots n_k} \approx \Z_{n_1} \oplus \Z_{n_2} \oplus \dots \oplus \Z_{n_k}}$]
|
||||
Let $m = n_1n_2\dots n_k$. Then $\Z_m$ is isomorphic to $\Z_{n_1} \oplus \Z_{n_2} \oplus \dots \oplus \Z_{n_k}$ if and only if $n_i$ and $n_j$ are relatively prime when $i \neq j$.
|
||||
\end{corollary}
|
||||
+17
@@ -0,0 +1,17 @@
|
||||
\section{The Group of Units Modulo $\mathbf{n}$ as an External Direct Product}
|
||||
|
||||
\begin{remark}
|
||||
The $U$-groups provide a convenient way to illustrate the preceding ideas. We first introduce some notation. If $k$ is a divisor of $n$, let
|
||||
\[ U_k(n) = \{x \in U(n)\ \vert\ x \mod k = 1\} \]
|
||||
\end{remark}
|
||||
|
||||
\begin{theorem}[$\mathbf{U(n)}$ as an External Direct Product]
|
||||
Suppose $s$ and $t$ are relatively prime. Then $U(st)$ is isomorphic to the external direct product of $U(s)$ and $U(t)$. In short,
|
||||
\[ U(st) \approx U(s) \oplus U(t) \]
|
||||
Moreover, $U_s(st)$ is isomorphic to $U(t)$ and $U_t(st)$ is isomorphic to $U(s)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}
|
||||
Let $m = n_1n_2\dots n_k$, where $\gcd(n_i,n_j)=1$ for $i \neq j$. Then,
|
||||
\[ U(m) \approx U(n_1) \oplus U(n_2) \oplus \dots \oplus U(n_k) \]
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Applications of Factor Groups}
|
||||
|
||||
\begin{theorem}[$\mathbf{G/Z}$ Theorem]
|
||||
Let $G$ be a group and let $Z(G)$ be the center of $G$. If $G/Z(G)$ is cyclic, then $G$ is Abelian.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[$\mathbf{G/Z(G) \approx \text{Inn}(G)}$]
|
||||
For any group $G$, $G/Z(G)$ is isomorphic to Inn$(G)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Cauchy's Theorem for Abelian Groups]
|
||||
Let $G$ be a finite Abelian group and let $p$ be a prime that divides the order of $G$. Then $G$ has an element of order $p$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,5 @@
|
||||
\chapter{Normal Subgroups and Factor Groups}
|
||||
\subimport{./}{normal-subgroups.tex}
|
||||
\subimport{./}{factor-groups.tex}
|
||||
\subimport{./}{applications-of-factor-groups.tex}
|
||||
\subimport{./}{internal-direct-products.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Factor Groups}
|
||||
|
||||
\begin{theorem}[Factor Groups (O. Hölder, 1889)]
|
||||
Let $G$ be a group and let $H$ be a normal subgroup of $G$. The set $G/H = \{ aH\ \vert\ a \in G\}$ is a group under the operation $(aH)(bH) = abH$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,26 @@
|
||||
\section{Internal Direct Products}
|
||||
|
||||
\begin{definition}[Internal Direct Product of $\mathbf{H}$ and $\mathbf{K}$]
|
||||
We say that $G$ is the \textit{internal direct product} of $H$ and $K$ and write $G = H \times K$ if $H$ and $K$ are normal subgroups of $G$ and
|
||||
\[ G = HK\ \ \ \ \text{and}\ \ \ \ H \cap K = \{e\} \]
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Internal Direct Product $\mathbf{H_1 \times H_2 \times \dots \times H_n}$]
|
||||
Let $H_1, H_2,\dots,H_n$ be a finite collection of normal subgroups of $G$. We say that $G$ is the \textit{internal direct product} of $H_1,H_2,\dots,H_n$ and write $G=H_1\times H_2 \times \dots \times H_n$, if
|
||||
\begin{enumerate}
|
||||
\item $G = H_1H_2\dots H_n = \{h_1h_2\dots h_n\ \vert\ h_i \in H_i\}$,
|
||||
\item $(H_1H_2\dots H_n) \cap H_{i + 1} = {e}$ for $i=1,2,\dots, n-1$.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[$\mathbf{H_1 \times H_2 \times \dots \times H_n \approx H_1 \oplus H_2 \oplus \dots \oplus H_n}$]
|
||||
If a group $G$ is the internal direct product of a finite number of subgroups $H_1,H_2, \dots, H_n$, then $G$ is isomorphic to the external direct product of $H_1,H_2 \dots, H_n$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Classification of Groups of Order $\mathbf{p^2}$]
|
||||
Every group of order $p^2$, where $p$ is a prime, is isomorphic to $\Z_{p^2}$ or $\Z_p \oplus \Z_p$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}
|
||||
If $G$ is a group of order $p^2$, where $p$ is a prime, then $G$ is Abelian.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{Normal Subgroups}
|
||||
|
||||
\begin{definition}[Normal Subgroup]
|
||||
A subgroup $H$ of a group $G$ is called a \textit{normal} subgroup of $G$ if $aH = Ha$ for all $a$ in $G$. We denote this by $H \triangleleft G$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Normal Subgroup Test]
|
||||
A subgroup $H$ of $G$ is normal in $G$ if and only if $xHx^{-1} \subseteq H$ for all $x$ in $G$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,12 @@
|
||||
\part{Groups}
|
||||
\setcounter{chapter}{1}
|
||||
\subimport{chapters/chapter-2/}{chapter-2.tex}
|
||||
\subimport{chapters/chapter-3/}{chapter-3.tex}
|
||||
\subimport{chapters/chapter-4/}{chapter-4.tex}
|
||||
\subimport{chapters/chapter-5/}{chapter-5.tex}
|
||||
\subimport{chapters/chapter-6/}{chapter-6.tex}
|
||||
\subimport{chapters/chapter-7/}{chapter-7.tex}
|
||||
\subimport{chapters/chapter-8/}{chapter-8.tex}
|
||||
\subimport{chapters/chapter-9/}{chapter-9.tex}
|
||||
\subimport{chapters/chapter-10/}{chapter-10.tex}
|
||||
\subimport{chapters/chapter-11/}{chapter-11.tex}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Introduction to Rings}
|
||||
\subimport{./}{motivation-and-definition.tex}
|
||||
\subimport{./}{properties-of-rings.tex}
|
||||
\subimport{./}{subrings.tex}
|
||||
@@ -0,0 +1,20 @@
|
||||
\section{Motivation and Definition}
|
||||
|
||||
\begin{definition}[Ring]
|
||||
A \textit{ring} $R$ is a set with two binary operations, addition (denoted by $a + b$) and multiplication (denoted by $ab$), such that for all $a,b,c$ in $R$:
|
||||
\begin{enumerate}
|
||||
\item $a + b = b + a$.
|
||||
\item $(a + b) + c = a + (b + c)$.
|
||||
\item There is an additive identity 0. That is, there is an element 0 in $R$ such that $a + 0 = a$ for all $a$ in $R$.
|
||||
\item There is an element $-a$ in $R$ such that $a + (-a) = 0$.
|
||||
\item $a(bc) = (ab)c$.
|
||||
\item $a(b+c) = ab + ac$ and $(b + c)a = ba + ca$.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
Note that multiplication need not be commutative. When it is, we say that the ring is \textit{commutative}. Also, a ring need not have an identity under multiplication. A \textit{unity} (or \textit{identity}) in a ring is a nonzero element that is an identity under multiplication. A nonzero element of a com-
|
||||
mutative ring with unity need not have a multiplicative inverse. When it does, we say that it is a unit of the ring. Thus, $a$ is a unit if $a^{-1}$ exists.
|
||||
|
||||
\noindent The following terminology and notation are convenient. If $a$ and $b$ belong to a commutative ring $R$ and $a$ is nonzero, we say that $a$ \textit{divides} $b$ (or that $a$ is a \textit{factor} of $b$) and write $a \vert b$, if there exists an element $c$ in $R$ such that $b = ac$. If $a$ does not divide $b$, we write $a \nmid b$.
|
||||
\end{remark}
|
||||
@@ -0,0 +1,23 @@
|
||||
\section{Properties of Rings}
|
||||
|
||||
\begin{theorem}[Rules of Multiplication]
|
||||
Let $a,b$, and $c$ belong to a ring $R$. Then
|
||||
\begin{enumerate}
|
||||
\item $a0 = 0a = 0$.
|
||||
\item $a(-b) = (-a)b = -(ab)$.
|
||||
\item $(-a)(-b) = ab$.
|
||||
\item $a(b-c) = ab - ac$ and $(b-c)a = ba - ca$.
|
||||
\end{enumerate}
|
||||
|
||||
Furthermore, if $R$ has a unity element $1$, then
|
||||
|
||||
\begin{enumerate}
|
||||
\setcounter{enumi}{4}
|
||||
\item $(-1)a = -a$.
|
||||
\item $(-1)(-1) = 1$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Uniqueness of the Unity and Inverses]
|
||||
If a ring has a unity, it is unique. If a ring element has a multiplicative inverse, it is unique.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{Subrings}
|
||||
|
||||
\begin{definition}[Subring]
|
||||
A subset $S$ of a ring $R$ is a \textit{subring of $R$} if $S$ is itself a ring with the operations of $R$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Subring Test]
|
||||
A nonempty subset $S$ of a ring $R$ is a subring if $S$ is closed under subtraction and multiplication -- that is, if $a - b$ and $ab$ are in $S$ whenever $a$ and $b$ are in $S$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Integral Domains}
|
||||
\subimport{./}{definition-and-examples.tex}
|
||||
\subimport{./}{fields.tex}
|
||||
\subimport{./}{characteristic-of-a-ring.tex}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Characteristic of a Ring}
|
||||
|
||||
\begin{definition}[Characteristic of a Ring]
|
||||
The \textit{characteristic} of a ring $R$ is the least positive integer $n$ such that $nx = 0$ for all $x$ in $R$. If no such integer exists, we say that $R$ has characteristic 0. The characteristic of $R$ is denoted by $\characteristic R$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Characteristic of a Ring with Unity]
|
||||
Let $R$ be a ring with unity 1. If 1 has infinite order under addition, then the characteristic of $R$ is 0. If 1 has order $n$ under addition, then the characteristic of $R$ is $n$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Characteristic of an Integral Domain]
|
||||
The characteristic of an integral domain is 0 or prime.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Definition and Examples}
|
||||
|
||||
\begin{definition}[Zero Divisors]
|
||||
A \textit{zero-divisor} is a nonzero element $a$ of a commutative ring $R$ such that there is a nonzero element $b \in R$ with $ab = 0$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Integral Domain]
|
||||
An \textit{integral domain} is a commutative ring with unity and no zero-divisors.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Cancellation]
|
||||
Let $a,b$, and $c$ belong to an integral domain If $a \neq 0$ and $ab = ac$, then $b = c$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Fields}
|
||||
|
||||
\begin{definition}[Field]
|
||||
A \textit{field} is a commutative ring with unity in which every nonzero element is a unit.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Finite Integral Domains are Fields]
|
||||
A finite integral domain is a field.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[$\mathbf{\Z_p}$ Is a Field]
|
||||
For every prime $p$, $\Z_p$, the ring of integers modulo $p$ is a field.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Ideals and Factor Rings}
|
||||
\subimport{./}{ideals.tex}
|
||||
\subimport{./}{factor-rings.tex}
|
||||
\subimport{./}{prime-ideals-and-maximal-ideals.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Factor Rings}
|
||||
|
||||
\begin{theorem}[Existence of Factor Rings]
|
||||
Let $R$ be a ring and let $A$ be a subring of $R$. The set of cosets $\{r + A\ \vert\ r \in R\}$ is a ring under the operations $(s + A) + (t + A) = s + t + A$ and $(s+A)(t+A)=st+A$ if and only if $A$ is an ideal of $R$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Ideals}
|
||||
|
||||
\begin{definition}[Ideal]
|
||||
A subring $A$ of a ring $R$ is called a (two-sided) \textit{ideal} of $R$ if for every $r \in R$ and every $a \in A$ both $ra$ and $ar$ are in $A$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Ideal Test]
|
||||
A nonempty subset $A$ of a ring $R$ is an ideal of $R$ if
|
||||
\begin{enumerate}
|
||||
\item $a-b \in A$ whenever $a,b \in A$.
|
||||
\item $ra$ and $ar$ are in $A$ whenever $a \in A$ and $r \in R$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,17 @@
|
||||
\section{Prime Ideals and Maximal Ideals}
|
||||
|
||||
\begin{remark}
|
||||
A \textit{proper} ideal is an ideal $I$ of some ring $R$ such that it is a proper subset of $R$; that is, $I \subset R$.
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}[Prime Ideal, Maximal Ideal]
|
||||
A \textit{prime ideal} $A$ of a commutative ring $R$ is a proper ideal of $R$ such that $a,b \in R$ and $ab \in A$ imply $a \in A$ or $b \in A$. A \textit{maximal} ideal of a commutative ring $R$ is a \textit{proper} ideal of $R$ such that, whenever $B$ is an ideal of $R$ and $A \subseteq B \subseteq R$, then $B = A$ or $B = R$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[$\mathbf{R/A}$ Is an Integral Domain If and Only If $\mathbf{A}$ Is Prime]
|
||||
Let $R$ be a commutative ring with unity and let $A$ be an ideal of $R$. Then $R/A$ is an integral domain if and only if $A$ is prime.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[$\mathbf{R/A}$ Is a Field If and Only If $\mathbf{A}$ Is Maximal]
|
||||
Let $R$ be a commutative ring with unity and let $A$ be an ideal of $R$. Then $R/A$ is a field if and only if $A$ is maximal.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Ring Homomorphisms}
|
||||
\subimport{./}{definition-and-examples.tex}
|
||||
\subimport{./}{properties-of-ring-homomorphisms.tex}
|
||||
\subimport{./}{the-field-of-quotients.tex}
|
||||
@@ -0,0 +1,7 @@
|
||||
\section{Definition and Examples}
|
||||
|
||||
\begin{definition}[Ring Homomorphism, Ring Isomorphism]
|
||||
A \textit{ring homomorphism} $\phi$ from a ring $R$ to a ring $S$ is a mapping from $R$ to $S$ that preserves the two ring operations; that is, for all $a,b$ in $R$,
|
||||
\[ \phi(a + b) = \phi(a) + \phi(b)\ \ \ \ \text{and}\ \ \ \ \phi(ab) = \phi(a)\phi(b) \]
|
||||
A ring homomorphism that is both one-to-one and onto is called a \textit{ring isomorphism}.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,42 @@
|
||||
\section{Properties of Ring Homomorphisms}
|
||||
|
||||
\begin{theorem}[Properties of Ring Homomorphisms]
|
||||
Let $\phi$ be a ring homomorphism from a ring $R$ to a ring $S$. Let $A$ be a subring of $R$ and let $B$ be an ideal of $S$.
|
||||
\begin{enumerate}
|
||||
\item For any $r \in R$ and any positive integer $n$, $\phi(nr) = n\phi(r)$ and $\phi(r^n) = (\phi(r))^n$.
|
||||
\item $\phi(A) = \{\phi(a)\ \vert\ a \in A\}$ is a subring of $S$.
|
||||
\item If $A$ is an ideal and $\phi$ is onto $S$, then $\phi(A)$ is an ideal.
|
||||
\item $\phi^{-1}(B) = \{r \in R\ \vert\ \phi(r) \in B\}$ is an ideal of $R$.
|
||||
\item If $R$ is commutative, then $\phi(R)$ is commutative.
|
||||
\item If $R$ has a unity 1, $S \neq \{0\}$, and $\phi$ is onto, then $\phi(1)$ is the unity of $S$.
|
||||
\item $\phi$ is an isomorphism if and only if $\phi$ is onto and $\ker \phi = \{r \in R\ \vert\ \phi(r) = 0\} = \{0\}$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Kernels Are Ideals]
|
||||
Let $\phi$ be a ring homomorphism from a ring $R$ to a ring $S$. Then $\ker \phi = \{r \in R\ \vert\ \phi(r) = 0\}$ is an ideal of $R$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[First Isomorphism Theorem for Rings]
|
||||
Let $\phi$ be a ring homomorphism from $R$ to $S$. Then the mapping from $R/\ker \phi$ to $\phi(R)$, given by $r + \ker \phi \to \phi(r)$, is an isomorphism. In symbols, $R/\ker\phi\approx\phi(R)$. This theorem is often referred to as the \textit{Fundamental Theorem of Ring Homomorphisms}.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Ideals Are Kernels]
|
||||
Every ideal of a ring $R$ is the kernel of a ring homomorphism of $R$. In particular, an idea l$A$ is the kernel of the mapping $r \to r + A$ from $R$ to $R/A$. This mapping is known as the \textit{natural homomorphism} from $R$ to $R/A$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Homomorphism from $\mathbf{\Z}$ to a Ring with Unity]
|
||||
Let $R$ be a ring with unity 1. The mapping $\phi: \Z \to R$ given by $n \to n \cdot 1$ is a ring homomorphism.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[A Ring with Unity Contains $\mathbf{\Z_n}$ or $\mathbf{\Z}$]
|
||||
If $R$ is a ring with unity and the characteristic of $R$ is $n > 0$, then $R$ contains a subring isomorphic to $\Z_n$. If the characteristic of $R$ is 0, then $R$ contains a subring isomorphic to $\Z$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[$\mathbf{\Z_m}$ Is a Homomorphic Image of $\mathbf{\Z}$]
|
||||
For any positive integer $m$, the mapping of $\phi: \Z \to \Z_m$ given by $x \to x \mod m$ is a ring homomorphism.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[A Field Contains $\mathbf{\Z_p \text{ or } \Q}$ (Steinitz, 1910)]
|
||||
If $\F$ is a field of characteristic $p$, then $\F$ contains a subfield isomorphic to $\Z_p$. If $\F$ is a field of characteristic 0, then $\F$ contains a subfield isomorphic to the rational numbers.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{The Field of Quotients}
|
||||
|
||||
\begin{theorem}[Field of Quotients]
|
||||
Let $D$ be an integral domain. Then there exists a field $\F$ (called the field of quotients in $D$) that contains a subring isomorphic to $D$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,3 @@
|
||||
\chapter{Polynomial Rings}
|
||||
\subimport{./}{notation-and-terminology.tex}
|
||||
\subimport{./}{the-division-algorithm-and-consequences.tex}
|
||||
@@ -0,0 +1,30 @@
|
||||
\section{Notation and Terminology}
|
||||
|
||||
\begin{definition}[Ring of Polynomials over $\mathbf{R}$]
|
||||
Let $R$ be a commutative ring. The set of formal symbols
|
||||
\[ R[x] = \{a_nx^n + a_{n-1}x^{n-1}+\dots+a_1x + a_0\ \vert\ a_i \in R, n \in \Z^+\} \]
|
||||
is called the \textit{ring of polynomials over $R$ in the indeterminate $x$}.\\
|
||||
\noindent Two elements
|
||||
\[ a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]
|
||||
\noindent and
|
||||
\[ b_mx^m + b_{m-1}x^{m-1} + \dots + b_1x + b_0 \]
|
||||
\noindent of $R[x]$ are considered equal if and only if $a_i=b_i$ for all nonnegative integers $i$. (Define $a_i=0$ when $i > n$ and $b_i = 0$ when $i > m$.)
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Addition and Multiplication in $\mathbf{R[x]}$]
|
||||
Let $R$ be a commutative ring and let
|
||||
\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]
|
||||
\noindent and
|
||||
\[ g(x) = b_mx^m + b_{m-1}x^{m-1} + \dots + b_1x + b_0 \]
|
||||
\noindent belong to $R[x]$. Then
|
||||
\[ f(x) + g(x) = (a_s + b_s)x^s + (a_{s-1} + b_{s-1})x^{s-1} + \dots + (a_1 + b_1)x + a_0 + b_0 \]
|
||||
\noindent where $s$ is the maximum of $m$ and $n$, $a_i = 0$ for $i > n$, and $b_i = 0$ for $i > m$. Also,
|
||||
\[ f(x)g(x) = c_{m+n}x^{m+n}+c_{m+n-1}x^{m+n-1} + \dots + c_1x + c_0 \]
|
||||
\noindent where
|
||||
\[ c_k = a_kb_0 + a_{k-1}b_1 + \dots + a_1b_{k-1} + a_0b_k \]
|
||||
\noindent for $k=0,\dots, m+n$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[$\mathbf{D}$ an Integral Domain Implies $\mathbf{D[x]}$ an Integral Domain]
|
||||
If $D$ is an integral domain, then $D[x]$ is an integral domain.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,29 @@
|
||||
\section{The Division Algorithm and Consequences}
|
||||
|
||||
\begin{theorem}[Division Algorithm for $\mathbf{\F[x]}$]
|
||||
Let $\F$ be a field and let $f(x), g(x) \in \F[x]$ with $g(x) \neq 0$. Then there exist unique polynomials $q(x)$ and $r(x)$ in $\F[x]$ such that $f(x) = g(x)q(x) + r(x)$ and either $r(x) = 0$ or $\deg r(x) < \deg g(x)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Remainder Theorem]
|
||||
Let $\F$ be a field, $a \in \F$, and $f(x) \in \F[x]$. Then $f(a)$ is the remainder in the division of $f(x)$ by $x -a$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Factor Theorem]
|
||||
Let $\F$ be a field, $a \in \F$, and $f(x) \in \F[x]$. Then $a$ is a zero of $f(x)$ if and only if $x-a$ is a factor of $f(x)$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[Polynomials of Degree $\mathbf{n}$ Have at Most $\mathbf{n}$ Zeros]
|
||||
A polynomial of degree $n$ over a field has at most $n$ zeros, counting multiplicity.
|
||||
\end{corollary}
|
||||
|
||||
\begin{definition}[Principal Ideal Domain (PID)]
|
||||
A \textit{principal ideal domain} is an integral domain $R$ in which every ideal has the form $\lr{a}=\{ra\ \vert\ r \in R\}$ for some $a$ in $R$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[$\mathbf{\F[x]}$ Is a PID]
|
||||
Let $\F$ be a field. Then $\F[x]$ is a principal ideal domain.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Criterion for $\mathbf{I = \lr{g(x)}}$]
|
||||
Let $\F$ be a field, $I$ a nonzero ideal in $\F[x]$, and $g(x)$ an element of $\F[x]$. Then, $I=\lr{g(x)}$ if and only if $g(x)$ is a nonzero polynomial of minimum degree in $I$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Factorization of Polynomials}
|
||||
\subimport{./}{reducibility-tests.tex}
|
||||
\subimport{./}{irreducibility-tests.tex}
|
||||
\subimport{./}{unique-factorization-in-zx.tex}
|
||||
@@ -0,0 +1,29 @@
|
||||
\section{Irreducibility Tests}
|
||||
|
||||
\begin{theorem}[Mod $\mathbf{p}$ Irreducibility Test]
|
||||
Let $p$ be a prime and suppose that $f(x) \in \Z[x]$ with $\deg f(x) \geq 1$. Let $\overline{f}(x)$ be the polynomial in $\Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ modulo $p$. If $\overline{f}(x)$ is irreducible over $\Z_p$ and $\deg \overline{f}(x) = \deg f(x)$, then $f(x)$ is irreducible over $\Q$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Eisenstein's Criterion (1850)]
|
||||
Let
|
||||
\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_0 \in \Z[x] \]
|
||||
\noindent If there is a prime $p$ such that $p \nmid a_n, p\ \vert\ a_{n-1}, \dots, p\ \vert\ a_0$ and $p^2 \nmid a_0$, then $f(x)$ is irreducible over $\Q$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Irreducibility of $\mathbf{p}$th Cyclotomic Polynomial]
|
||||
For any prime $p$, the $p$th cyclotomic polynomial
|
||||
\[ \Phi_p(x) = \frac{x^p - 1}{x-1} = x^{p-1} + x^{p-2} + \dots + x + 1 \]
|
||||
\noindent is irreducible over $\Q$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[$\mathbf{\lr{p(x)}}$ Is Maximal If and Only If $\mathbf{p(x)}$ Is Irreducible]
|
||||
Let $\F$ be a field and let $p(x) \in \F[x]$. Then $\lr{p(x)}$ is a maximal ideal in $\F[x]$ if and only if $p(x)$ is irreducible over $\F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[$\mathbf{\F[x]/\lr{p(x)}}$ Is a Field]
|
||||
Let $\F$ be a field and $p(x)$ be an irreducible polynomial over $\F$. Then $\F[x]/\lr{p(x)}$ is a field.
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[$\mathbf{p(x)\ \vert\ a(x)b(x)}$ Implies $\mathbf{p(x)\ \vert\ a(x)}$ or $\mathbf{p(x)\ \vert\ b(x)}$]
|
||||
Let $\F$ be a field and let $p(x), a(x), b(x) \in \F[x]$. If $p(x)$ is irreducible over $\F$ and $p(x)\ \vert\ a(x)b(x)$, then $p(x)\ \vert\ a(x)$ or $p(x)\ \vert\ b(x)$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,21 @@
|
||||
\section{Reducibility Tests}
|
||||
|
||||
\begin{definition}[Irreducible Polynomial, Reducible Polynomial]
|
||||
Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be \textit{irreducible over $D$}, whenever $f(x)$ is expressed as a product $f(x) = g(x)h(x)$, with $g(x)$ and $h(x)$ from $D[x]$, then $g(x)$ or $h(x)$ is a unit in $D[x]$. A nonzero, nonunit element of $D[x]$ that is not irreducible over $D$ is called \textit{reducible over $D$}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Reducibility Test for Degrees 2 and 3]
|
||||
Let $\F$ be a field. If $f(x) \in \F[x]$ and $\deg f(x)$ is 2 or 3, then $f(x)$ is reducible over $\F$ if and only if $f(x)$ has a zero in $\F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Content of a Polynomial, Primitive Polynomial]
|
||||
The \textit{content} of a nonzero polynomial $a_nx^n + a_{n-1}x^{n-1} + \dots + a_0$, where the $a$'a are integers, is the greatest common divisor of the integers $a_n,a_{n-1}, \dots, a_0$. A \textit{primitive polynomial} is an element of $\Z[x]$ with content 1.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}[Gauss's Lemma]
|
||||
The product of two primitive polynomials is primitive.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[Reducibility over $\mathbf{\Q}$ Implies Reducibility over $\mathbf{\Z}$]
|
||||
Let $f(x) \in \Z[x]$. If $f(x)$ is reducible over $\Q$, then it is reducible over $\Z$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,7 @@
|
||||
\section{Unique Factorization In $\mathbf{\Z[x]}$}
|
||||
|
||||
\begin{theorem}[Unique Factorization in $\mathbf{\Z[x]}$]
|
||||
Every polynomial in $\Z[x]$ that is not the zero polynomial or a unit in $\Z[x]$ can be written in the form $b_1b_2\dots b_sp_1(x)p_2(x)\dots p_m(x)$, where the $b_i$'s are irreducible polynomials of degree 0 and the $p_i(x)$'s are irreducible polynomials of positive degree. Furthermore, if
|
||||
\[ b_1b_2\dots b_sp_1(x)p_2(x) \dots p_m(x) = c_1c_2 \dots c_tq_1(x) q_2(x) \dots q_n(x) \]
|
||||
\noindent where the $b_i$'s and the $c_i$'s are irreducible polynomials of degree 0 and the $p_i(x)$'s and $q_i(x)$'s are irreducible polynomials of positive degree, then $s=t, m=n$, and, after renumbering the $c$'s and $q(x)$'s, we have $b_i = \pm c_i$, for $i=1, \dots, s$, and $p_i(x)= \pm q_i(x)$, for $i = 1, \dots, m$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Divisibility in Integral Domains}
|
||||
\subimport{./}{irreducibles-primes.tex}
|
||||
\subimport{./}{unique-factorization-domains.tex}
|
||||
\subimport{./}{euclidean-domains.tex}
|
||||
@@ -0,0 +1,21 @@
|
||||
\section{Euclidean Domains}
|
||||
|
||||
\begin{definition}[Euclidean Domain (ED)]
|
||||
An integral domain $D$ is called a \textit{Euclidean domain} if there is a function $d$ (called the \textit{measure}) from nonzero elements of $D$ to the nonnegative integers such that
|
||||
\begin{enumerate}
|
||||
\item $d(a) \leq d(ab)$ for all nonzero $a,b \in D$; and
|
||||
\item if $a,b \in D,\ b \neq 0$, then there exist elements $q$ and $r$ in $D$ such that $a = bq + r$, where $r = 0$ or $d(r) < d(b)$.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[ED Implies PID]
|
||||
Every Euclidean domain is a principal ideal domain.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[ED Implies UFD]
|
||||
Every Euclidean domain is a unique factorization domain.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[$\mathbf{D}$ a UFD Implies $\mathbf{D[x]}$ a UFD]
|
||||
If $D$ is a unique factorization domain, then $D[x]$ is a unique factorization domain.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,13 @@
|
||||
\section{Irreducibles, Primes}
|
||||
|
||||
\begin{definition}[Associates, Irreducibles, Primes]
|
||||
Elements $a$ and $b$ of an integral domain $D$ are called \textit{associates} if $a = ub$, where $u$ is a unit of $D$. A nonzero element $a$ of an integral domain $D$ is called an \textit{irreducible} if $a$ is not a unit and, whenever $b$, $c \in D$ with $a = bc$, then $b$ or $c$ is a unit. A nonzero element $a$ of an integral domain $D$ is called a \textit{prime} if $a$ is not a unit and $a\ \vert\ bc$ implies $a\ \vert\ b$ or $a\ \vert\ c$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Prime Implies Irreducible]
|
||||
In an integral domain, every prime in an irreducible.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[PID Implies Irreducible Equals Prime]
|
||||
In a principal ideal domain, an element is an irreducible if and only if it is a prime.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,21 @@
|
||||
\section{Unique Factorization Domains}
|
||||
|
||||
\begin{definition}
|
||||
An integral domain $D$ is a \textit{unique factorization domain} if
|
||||
\begin{enumerate}
|
||||
\item every nonzero element of $D$ that is not a unit can be written as a product of irreducibles of $D$; and
|
||||
\item the factorization into irreducibles is unique up to associates and the order in which the factors appear.
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}[Ascending Chain Condition for a PID]
|
||||
In a principal ideal domain, any stricly increasing chain of ideals $I_1 \subset I_2 \subset \dots$ must be finite in length.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[PID Implies UFD]
|
||||
Every principal ideal domain is a unique factorization domain.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[$\mathbf{\F[x]}$ Is a UFD]
|
||||
Let $\F$ be a field. Then $\F[x]$ is a unique factorization domain.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,8 @@
|
||||
\part{Rings}
|
||||
\subimport{chapters/chapter-12/}{chapter-12.tex}
|
||||
\subimport{chapters/chapter-13/}{chapter-13.tex}
|
||||
\subimport{chapters/chapter-14/}{chapter-14.tex}
|
||||
\subimport{chapters/chapter-15/}{chapter-15.tex}
|
||||
\subimport{chapters/chapter-16/}{chapter-16.tex}
|
||||
\subimport{chapters/chapter-17/}{chapter-17.tex}
|
||||
\subimport{chapters/chapter-18/}{chapter-18.tex}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Vector Spaces}
|
||||
\subimport{./}{definition-and-examples.tex}
|
||||
\subimport{./}{subspaces.tex}
|
||||
\subimport{./}{linear-independence.tex}
|
||||
@@ -0,0 +1,15 @@
|
||||
\section{Definition and Examples}
|
||||
|
||||
\begin{definition}[Vector Space]
|
||||
A set $V$ is said to be a \textit{vector space} over a field $\F$ if $V$ is an Abelian group under addition (denoted by $+$) and, if for each $a \in \F$ and $v \in V$, there is an element $av \in V$ such that the following conditions hold for all $a,b \in \F$ and all $u,v \in V$.
|
||||
\begin{enumerate}
|
||||
\item $a(v + u) = av + au$
|
||||
\item $(a + b)v = av + bv$
|
||||
\item $a(bv)=(ab)v$
|
||||
\item $1v=v$
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
The members of a vector space are called \textit{vectors}. The members of the field are called \textit{scalars}. The operation that combines a scalar $a$ and a vector $v$ to form the vector $av$ is called \textit{scalar multiplication}. In general, we will denote vectors by letters from the end of the alphabet, such as $u,v,w$, and scalars by letters from the beginning of the alphabet, such as $a,b,c$.
|
||||
\end{remark}
|
||||
@@ -0,0 +1,19 @@
|
||||
\section{Linear Independence}
|
||||
|
||||
\begin{definition}[Linearly Dependent, Linearly Independent]
|
||||
A set $S$ of vectors is said to be \textit{linearly dependent} over a field $\F$ if there are vectors $v_1,v_2,\dots,v_n$ from $S$ and elements $a_1,a_2,\dots,a_n$ from $\F$, not all zero, such that $a_1v_1+a_2v_2+\dots+a_nv_n = 0$. A set of vectors that is not linearly dependent over $\F$ is called \textit{linearly independent} over $\F$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}[Basis]
|
||||
Let $V$ be a vector space over $\F$. A subset $B$ of $V$ is called a \textit{basis} for $V$ if $B$ is linearly independent over $\F$ and every element of $V$ is a linear combination of elements of $B$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Invariance of Basis Size]
|
||||
If $\{u_1,u_2,\dots,u_m\}$ and $\{w_1,w_2,\dots,w_n\}$ are both bases of a vector space $V$ over a field $\F$, then $m=n$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Dimension]
|
||||
A vector space that has a basis consisting of $n$ elements is said to have \textit{dimension $n$}. For completeness, the trivial vector space $\{0\}$ is said to be spanned by the empty set and to have dimension 0.
|
||||
|
||||
\noindent A vector space that has a finite basis is called \textit{finite dimensional}; otherwise, it is called \textit{infinite dimensional}.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Subspaces}
|
||||
|
||||
\begin{definition}[Subspace]
|
||||
Let $V$ be a vector space over a field $\F$ and let $U$ be a subset of $V$. We say that $U$ is a \textit{subspace} of $V$ if $U$ is also a vector space over $\F$ under the operations of $V$.
|
||||
\end{definition}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Extension Fields}
|
||||
\subimport{./}{the-fundamental-theorem-of-field-theory.tex}
|
||||
\subimport{./}{splitting-fields.tex}
|
||||
\subimport{./}{zeros-of-an-irreducible-polynomial.tex}
|
||||
@@ -0,0 +1,34 @@
|
||||
\section{Splitting Fields}
|
||||
|
||||
\begin{definition}[Splitting Field]
|
||||
Let $\E$ be an extension field of $\F$ and let $f(x) \in \F[x]$ with degree at least 1. We say that $f(x)$ \textit{splits} in $\E$ if there are elements $a \in \F$ and $a_1,a_2,\dots,a_n \in \E$ such that
|
||||
\[ f(x) = a(x-a_1)(x-a_2)\dots(x-a_n) \]
|
||||
We call $\E$ a \textit{splitting field for $f(x)$ over $\F$} if
|
||||
\[ \E = \F(a_1,a_2,\dots,a_n) \]
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Existence of Splitting Fields]
|
||||
Let $\F$ be a field and let $f(x)$ be a nonconstant element of $\F[x]$. Then there exists a splitting field $\E$ for $f(x)$ over $\F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[$\mathbf{\F(a) \approx \F[x]/\lr{p(x)}}$]
|
||||
Let $\F$ be a field and let $p(x) \in \F[x]$ be irreducible over $\F$. If $a$ is a zero of $p(x)$ in some extension $\E$ of $\F$, then $\F(a)$ is isomorphic to $\F[x]/\lr{p(x)}$. Furthermore, if $\deg p(x) = n$, then every member of $\F(a)$ can be uniquely expressed in the form
|
||||
\[ c_{n-1}a^{n-1}+c_{n-2}a^{n-2}+\dots+c_1a+c_0 \]
|
||||
where $c_0,c_1,\dots,c_{n-1} \in \F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[$\mathbf{\F(a) \approx \F(b)}$]
|
||||
Let $\F$ be a field and let $p(x) \in \F[x]$ be irreducible over $\F$. If $a$ is a zero of $p(x)$ in some extension $\E$ of $\F$ and $b$ is a zero of $p(x)$ in some extension $\E'$ of $\F$, then the fields $\F(a)$ and $\F(b)$ are isomorphic.
|
||||
\end{corollary}
|
||||
|
||||
\begin{lemma}
|
||||
Let $\F$ be a field, let $p(x) \in \F[x]$ be irreducible over $\F$, and let $a$ be a zero of $p(x)$ in some extension of $\F$. If $\phi$ is a field isomorphism from $\F$ to $\F'$ and $b$ is a zero of $\phi(p(x))$ in some extension of $\F'$, then there is an isomorphism from $\F(a)$ to $\F'(b)$ that agrees with $\phi$ on $\F$ and carries $a$ to $b$.
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[Extending $\mathbf{\phi: \F \to \F'}$]
|
||||
Let $\phi$ be an isomorphism from a field $\F$ to a field $\F'$ and let $f(x) \in \F[x]$. If $\E$ is a splitting field for $f(x)$ over $\F$ and $\E'$ is a splitting field for $\phi(f(x))$ over $\F'$, then there is an isomorphism from $\E$ to $\E'$ that agrees with $\phi$ on $\F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Splitting Fields Are Unique]
|
||||
Let $\F$ be a field and let $f(x) \in \F[x]$. Then any two splitting fields of $f(x)$ over $\F$ are isomorphic.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{The Fundamental Theorem of Field Theory}
|
||||
|
||||
\begin{definition}[Extension Field]
|
||||
A field $\E$ is an \textit{extension field} of a field $\F$ if $\F \subseteq \E$ and the operations of $\F$ are those of $\E$ restricted to $\F$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Fundamental Theorem of Field Theory (Kronecker's Theorem, 1887)]
|
||||
Let $\F$ be a field and let $f(x)$ be a nonconstant polynomial in $\F[x]$. Then there is an extension field $\E$ of $\F$ in which $f(x)$ has a zero.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,44 @@
|
||||
\section{Zeros of an Irreducible Polynomial}
|
||||
|
||||
\begin{definition}[Derivative]
|
||||
Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$ belong to $\F[x]$. The \textit{derivative} of $f(x)$, denoted by $f'(x)$, is the polynomial $na_nx^{x-1} + (n-1)a_{n-1}x^{n-2} + \dots + a_1$ in $\F[x]$.
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}[Properties of the Derivative]
|
||||
Let $f(x)$ and $g(x) \in \F[x]$ and let $a \in \F$. Then
|
||||
\begin{enumerate}
|
||||
\item $(f(x) + g(x))' = f'(x) + g'(x)$.
|
||||
\item $(af(x))' = af'(x)$.
|
||||
\item $(f(x)g(x))' = f(x)g'(x) + g(x)f'(x)$.
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
|
||||
\begin{theorem}[Criterion for Multiple Zeros]
|
||||
A polynomial $f(x)$ over a field $\F$ has a multiple zero in some extension $\E$ if and only if $f(x)$ and $f'(x)$ have a common factor of positive degree in $\F[x]$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Zeros of an Irreducible]
|
||||
Let $f(x)$ be an irreducible polynomial over a field $\F$. If $\F$ has characteristic 0, then $f(x)$ has no multiple zeros. If $\F$ has characteristic $p \neq 0$, then $f(x)$ has a multiple zero if it is of the form $f(x) = g(x^p)$ for some $g(x)$ in $\F[x]$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}[Perfect Field]
|
||||
A field $\F$ is called \textit{perfect} if $\F$ has characteristic 0 or if $\F$ has characteristic $p$ and $\F^p=\{a^p\ \vert\ a \in \F\} = \F$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Finite Fields Are Perfect]
|
||||
Every finite field is perfect.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Criterion for No Multiple Zeros]
|
||||
If $f(x)$ is an irreducible polynomial over a perfect field $\F$, then $f(x)$ has no multiple zeros.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Zeros of an Irreducible over a Splitting Field]
|
||||
Let $f(x)$ be an irreducible polynomial over a field $\F$ and let $\E$ be a splitting field of $f(x)$ over $\F$. Then all the zeros of $f(x)$ in $\E$ have the same multiplicity.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Factorization of an Irreducible over a Splitting Field]
|
||||
Let $f(x)$ be an irreducible polynomial over a field $\F$ and let $\E$ be a splitting field of $f(x)$. Then $f(x)$ has the form
|
||||
\[ a(x-a_1)^n(x-a_2)^n\dots(x-a_t)^n \]
|
||||
where $a_1,a_2,\dots,a_t$ are distinct elements of $\E$ and $a \in \F$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Algebraic Extensions}
|
||||
\subimport{./}{characterization-of-extensions.tex}
|
||||
\subimport{./}{finite-extensions.tex}
|
||||
\subimport{./}{properties-of-algebraic-extensions.tex}
|
||||
@@ -0,0 +1,17 @@
|
||||
\section{Characterization of Extensions}
|
||||
|
||||
\begin{definition}[Types of Extensions]
|
||||
Let $\E$ be an extension field of a field $\F$ and let $a \in \E$. We call $a$ \textit{algebraic over $\F$} if $a$ is the zero of some nonzero polynomial in $\F[x]$. If $a$ is not algebraic over $\F$, it is called \textit{transcendental over $\F$}. An extension $\E$ of $\F$ is called an \textit{algebraic} extension of $\F$ if every element of $\E$ is algebraic over $\F$. If $\E$ is not an algebraic extension of $\F$, it is called a \textit{transcendental} extension of $\F$. An extension of $\F$ of the form $\F(a)$ is called a \textit{simple} extension of $\F$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Characterization of Extensions]
|
||||
Let $\E$ be an extension field of the field $\F$ and let $a \in \E$. If $a$ is transcendental over $\F$, then $\F(a) \approx \F(x)$. If $a$ is algebraic over $\F$, then $\F(a) \approx \F[x]/\lr{p(x)}$, where $p(x)$ is a polynomial in $\F[x]$ of minimum degree such that $p(a) = 0$. Moreover, $p(x)$ is irreducible over $\F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Uniqueness Property]
|
||||
If $a$ is algebraic over a field $\F$, then there is a unique monic irreducible polynomial $p(x)$ in $\F[x]$ such that $p(a)=0$. The polynomial with this property is called the \textit{minimal polynomial for $a$ over $\F$}.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Divisibility Property]
|
||||
Let $a$ be algebraic over $\F$, and let $p(x)$ be the minimal polynomial for $a$ over $\F$. If $f(x) \in \F[x]$ and $f(a) = 0$, then $p(x)$ divides $f(x)$ in $\F[x]$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,17 @@
|
||||
\section{Finite Extensions}
|
||||
|
||||
\begin{definition}[Degree of an Extension]
|
||||
Let $\E$ be an extension field of a field $\F$. We say that $\E$ \textit{has degree $n$ over $\F$} and write $[\E:\F]=n$ if $\E$ has dimension $n$ as a vector space over $\F$. If $[\E:\F]$ is finite, $\E$ is called a \textit{finite extension} of $\F$; otherwise, we say that $\E$ is an \textit{infinite extension} of $\F$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[Finite Implies Algebraic]
|
||||
If $\E$ is a finite extension of $\F$, then $\E$ is an algebraic extension of $\F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[$\mathbf{[\K:\F] = [\K:\E][\E:\F]}$]
|
||||
Let $\K$ be a finite extension field of the field $\E$ and let $\E$ be a finite extension field of the field $\F$. Then $\K$ is a finite extension field of $\F$ and $[\K:\F] = [\K:\E][\E:\F]$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}[Primitive Element Theorem (Steinitz, 1910)]
|
||||
If $\F$ is a field of characteristic 0, and $a$ and $b$ are algebraic over $\F$, then there is an element $c$ in $\F(a,b)$ such that $\F(a,b) = \F(c)$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,9 @@
|
||||
\section{Properties of Algebraic Extensions}
|
||||
|
||||
\begin{theorem}[Algebraic over Algebraic Is Algebraic]
|
||||
If $\K$ is an algebraic extension of $\E$ and $\E$ is an algebraic extension of $\F$, then $\K$ is an algebraic extension of $\F$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}[Subfield of Algebraic Elements]
|
||||
Let $\E$ be an extension field of the field $\F$. Then the set of all elements of $\E$ that are algebraic over $\F$ is a subfield of $\E$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,4 @@
|
||||
\chapter{Finite Fields}
|
||||
\subimport{./}{classification-of-finite-fields.tex}
|
||||
\subimport{./}{structure-of-finite-fields.tex}
|
||||
\subimport{./}{subfields-of-a-finite-field.tex}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Classification of Finite Fields}
|
||||
|
||||
\begin{theorem}[Classification of Finite Fields]
|
||||
For each prime $p$ and each positive integer $n$, there is, up to isomorphism, a unique finite field of order $p^n$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1,19 @@
|
||||
\section{Structure of Finite Fields}
|
||||
|
||||
\begin{theorem}[Structure of Finite Fields]
|
||||
As a group under addition, $\gf(p^n)$ is isomorphic to
|
||||
\[ \underbrace{\Z_p \oplus \Z_p \oplus \dots \oplus \Z_p}_\text{$n$ factors} \]
|
||||
As a group under multiplication, the set of nonzero elements of $\gf(p^n)$ is isomorphic to $\Z_{p^n-1}$ (and is, therefore, cyclic).
|
||||
\end{theorem}
|
||||
|
||||
\begin{remark}
|
||||
Because there is only one field for each prime-power $p^n$, we may unambiguously denote it by $\gf(p^n)$, in honor of Galois, and call it the \textit{Galois field of order $p^n$}.
|
||||
\end{remark}
|
||||
|
||||
\begin{corollary}
|
||||
\[ [\gf(p^n):\gf(p)]=n \]
|
||||
\end{corollary}
|
||||
|
||||
\begin{corollary}[$\mathbf{\gf(p^n)}$ Contains an Element of Degree $\mathbf{n}$]
|
||||
Let $a$ be a generator of the group of nonzero elements of $\gf(p^n)$ under multiplication. Then $a$ is algebraic over $\gf(p)$ of degree $n$.
|
||||
\end{corollary}
|
||||
@@ -0,0 +1,5 @@
|
||||
\section{Subfields of a Finite Field}
|
||||
|
||||
\begin{theorem}[Subfields of a Finite Field]
|
||||
For each divisor $m$ of $n$, $\gf(p^n)$ has a unique subfield of order $p^m$. Moreover, these are the only subfields of $\gf(p^n)$.
|
||||
\end{theorem}
|
||||
@@ -0,0 +1 @@
|
||||
\chapter{Geometric Constructions}
|
||||
@@ -0,0 +1,6 @@
|
||||
\part{Fields}
|
||||
\subimport{chapters/chapter-19/}{chapter-19.tex}
|
||||
\subimport{chapters/chapter-20/}{chapter-20.tex}
|
||||
\subimport{chapters/chapter-21/}{chapter-21.tex}
|
||||
\subimport{chapters/chapter-22/}{chapter-22.tex}
|
||||
\subimport{chapters/chapter-23/}{chapter-23.tex}
|
||||
Some files were not shown because too many files have changed in this diff Show More
Reference in New Issue
Block a user