Created the Abstract Algebra theorems and definitions cheat sheet

part-3/chapters/chapter-16/chapter-16.tex

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+\chapter{Polynomial Rings}
+\subimport{./}{notation-and-terminology.tex}
+\subimport{./}{the-division-algorithm-and-consequences.tex}

part-3/chapters/chapter-16/notation-and-terminology.tex

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+\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}

part-3/chapters/chapter-16/the-division-algorithm-and-consequences.tex

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+\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}