Exercise 10.1.7 (Verify field $\mathbb{F}_2(\alpha)$)

Question

Verify directly that $\beta^4 = \beta$ for all $\beta$ in the 4-element field $\mathbb{F}_2(\alpha)$ of Example 5.1.8.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
We have that $\mathbb{F}_2(\alpha) = \{0, 1, \alpha, 1+\alpha\}$. Then we have that:
\begin{equation*}
    \begin{aligned}
        0^4 &= 0\
        1^4 &= 1\
        \alpha^4 &= \alpha^2\alpha^2 = (1+\alpha)(1+\alpha) = \alpha\
        (1+\alpha)^4 &= 1^4 + \alpha^4 = 1 + \alpha \
    \end{aligned}
\end{equation*}
Indeed $\beta^4 = \beta$.

Exercise 10.1.7 (Verify field $\mathbb{F}_2(\alpha)$)

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Exercise 10.1.7 (Verify field $\mathbb{F}_2(\alpha)$)

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