Let $f=x^3-3x-1 \in \mathbb{Q}[X]$ and $a \in \mathbb{C}$ a root of $f$.
If we apply the Eisenstein's criterion for $g=f(x+1)$ we deduce that $f$ is irreducible over $\mathbb{Q}$.
Also $2-a^2$ and $a^2-a-2$ are roots of $f$ and the set $\{1,a,a^2\}$ forms a basis of the extension $\mathbb{Q}(a)/\mathbb{Q}$.
By this we deduce (proving with induction) that $\forall n \in \mathbb{N}$
$\exists c_0,c_1,c_2 \in \mathbb{Z} $ such that $(3+a-a^2)^n=c_0+c_1a+c_2a^2$
Now i've been asked to prove that also $(1-a)^n=(c_0+2c_1+4c_2)+c_2a-(c_1+c_2)a^2$ $(*)$
I proved that the extension is normal but i don't think that helps to prove $(*)$. I also tried to connect the previous relation with the $c_i$'s but i'm stuck for ideas.
Can someone help me please?
Keeping the notation of Jyrki Lahtonen we express the automorphism $\sigma$ as follows (using the basis $1,a,a^2$): $$\sigma(1) = 1$$ $$ \sigma(a) = a^2 -a-2$$ $$ \sigma(a^2)=(\sigma(a))^2=(a^2 -a-2)^2 = a^4-2a^3-3a^2+4a+4 = -a + 2$$ so that $\sigma$ can be expressed by the matrix $S = \begin{pmatrix} 1 & -2 & 2 \\ 0 & -1 & -1 \\ 0 & 1& 0\end{pmatrix}$ (note that $S^3 = 1$). We need to calculate $\sigma^2((3+a-a^2))=\sigma^2(c_0+c_1a+c_2a^2)$, or in matrix form: $$ S^2 \begin{pmatrix} c0 \\c1 \\ c2 \end{pmatrix} = \begin{pmatrix} c_0+2c_1+4c_2\\c_2 \\-c_1-c_2 \end{pmatrix} $$