I would be thankful if you could help me : I have studied many things about Galois fields, but now I am not sure about my understanding of frobenius maps. For example can anyone help me the matirx of Frobenius map on GF(8) as a vector space over GF(2)?
2026-03-30 08:55:51.1774860951
matrix for frobenius map of finite fields
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Realising $\Bbb{F}_8$ as $\Bbb{F}_2(\alpha)$ where $\alpha$ is a root of $X^3+X+1$, we have a basis $(1,\alpha,\alpha^2)$ for $\Bbb{F}_8$ over $\Bbb{F}_2$. With respect to this basis the Frobenius map on $\Bbb{F}_8$ is represented by the matrix $$\left[\begin{array}{ccc}1&0&0\\0&0&1\\0&1&1\end{array}\right].$$ Note that this is very much dependent on the choice of basis for $\Bbb{F}_8$ over $\Bbb{F}_2$.