49 lines
1.4 KiB
TeX
49 lines
1.4 KiB
TeX
\section{Fields}
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\definition{Fields and Field Axioms}
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A \textit{field} $\mathbb{F}$ consists of a set $A$ and two operations $+$ and $\times$. \\
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As usual, we may abbreviate $a \times b$ as $ab$. \\
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The following axioms must be satisfied for any $a, b, c \in \mathbb{F}$:
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\vspace{1mm}
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\begin{center}
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% @{} supresses the space between columns.
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% @{=} makes = a column seperator.
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\begin{tabular}{l | r@{=}l | r@{=}l}
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\hline
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\multicolumn{1}{|c|}{Name} &
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\multicolumn{2}{c}{$+$} &
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\multicolumn{2}{|c|}{$\times$} \\
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\hline
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Closure & \multicolumn{2}{c|}{$a+b \in \mathbb{F}$} & \multicolumn{2}{c}{$ab \in \mathbb{F}$} \\
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Associativity & $(a+b)+c~$&$~a+b+c$ & $(ab)c~$&$~a(bc)$ \\
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Commutativity & $a+b~$&$~b+a$ & $ab~$&$~ba$ \\
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Distributivity & $a(b+c)~$&$~ab + ac$ & \multicolumn{2}{}{} \\
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Identity & $a+0~$&$~a$ & $1 \times a~$&$~a$ \\
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Inverses & $a + (-a)~$&$~0$ & $a \times a^{-1}~$&$~1$
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\end{tabular}
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\end{center}
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\problem{}
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Show that all fields are groups. \\
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Convince yourself that not all groups are fields.
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\vfill
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\problem{}
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Is $\mathbb{Z}$ a field under our usual definitions of $+$ and $\times$? \\
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Which axioms does it satisfy, and which does it violate?
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\vfill
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\problem{}
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Verify that $\mathbb{R}$ is a field.
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\vfill
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\generic{Remark:}
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We won't worry too much about fields this week. They simply provide a foundation for \textit{spaces}. \\
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As such, you may assume that we are working in $\mathbb{R}$ for the rest of this handout.
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\pagebreak |