31 lines
1.5 KiB
TeX
31 lines
1.5 KiB
TeX
\section{Notation and Terminology}
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\begin{definition}[Ring of Polynomials over $\mathbf{R}$]
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Let $R$ be a commutative ring. The set of formal symbols
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\[ R[x] = \{a_nx^n + a_{n-1}x^{n-1}+\dots+a_1x + a_0\ \vert\ a_i \in R, n \in \Z^+\} \]
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is called the \textit{ring of polynomials over $R$ in the indeterminate $x$}.\\
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\noindent Two elements
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\[ a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]
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\noindent and
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\[ b_mx^m + b_{m-1}x^{m-1} + \dots + b_1x + b_0 \]
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\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 $i > m$.)
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\end{definition}
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\begin{definition}[Addition and Multiplication in $\mathbf{R[x]}$]
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Let $R$ be a commutative ring and let
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\[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]
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\noindent and
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\[ g(x) = b_mx^m + b_{m-1}x^{m-1} + \dots + b_1x + b_0 \]
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\noindent belong to $R[x]$. Then
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\[ f(x) + g(x) = (a_s + b_s)x^s + (a_{s-1} + b_{s-1})x^{s-1} + \dots + (a_1 + b_1)x + a_0 + b_0 \]
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\noindent where $s$ is the maximum of $m$ and $n$, $a_i = 0$ for $i > n$, and $b_i = 0$ for $i > m$. Also,
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\[ f(x)g(x) = c_{m+n}x^{m+n}+c_{m+n-1}x^{m+n-1} + \dots + c_1x + c_0 \]
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\noindent where
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\[ c_k = a_kb_0 + a_{k-1}b_1 + \dots + a_1b_{k-1} + a_0b_k \]
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\noindent for $k=0,\dots, m+n$.
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\end{definition}
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\begin{theorem}[$\mathbf{D}$ an Integral Domain Implies $\mathbf{D[x]}$ an Integral Domain]
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If $D$ is an integral domain, then $D[x]$ is an integral domain.
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\end{theorem}
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