Exercise 5.1.9 (Tables)

Question

Write out the addition and multiplication tables of $\mathbb{F}_2(\alpha)$.

Hassaan Naeem · Accepted Answer

ewcolumntype{2}{D{.}{}{4.0}}
	extbf{Solution:}
    \begin{center}
        enewcommand\arraystretch{1.3}
        \setlength\doublerulesep{0pt}
        \begin{tabular}{r||*{4}{2|}}
            + & 0 & 1 & \alpha & 1+\alpha \
            \hline\hline
            0 & 0 & 1 & \alpha & 1+\alpha \ 
            \hline
            1 & 1 & 0 & 1+\alpha & \alpha \ 
            \hline
            $\alpha$ & \alpha & 1+\alpha & 0 & 1 \ 
            \hline
            $1+\alpha$ & 1+\alpha & \alpha & 1 & 0 \ 
            \hline
        \end{tabular}
    \end{center}
    \vspace{0.5 cm}
    \begin{center}
        enewcommand\arraystretch{1.3}
        \setlength\doublerulesep{0pt}
        \begin{tabular}{r||*{4}{2|}}
            $	imes$ & 0 & 1 & \alpha & 1+\alpha \
            \hline\hline
            0 & 0 & 0 & 0 & 0 \ 
            \hline
            1 & 0 & 1 & \alpha & 1+\alpha \ 
            \hline
            $\alpha$ & 0 & \alpha & 1+\alpha & 1\ 
            \hline
            $1+\alpha$ & 0 & 1+\alpha & 1 & \alpha \ 
            \hline
        \end{tabular}
    \end{center}

+	0	1	α	1 + α


0	0	1	α	1 + α

1	1	0	1 + α	α

$α$	α	1 + α	0	1

$1 + α$	1 + α	α	1	0

$×$	0	1	α	1 + α


0	0	0	0	0

1	0	1	α	1 + α

$α$	0	α	1 + α	1

$1 + α$	0	1 + α	1	α

Exercise 5.1.9 (Tables)

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Exercise 5.1.9 (Tables)

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