\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}
