# Find c_1,c_2,c_3 in QQ such that (1+alpha^4)−1=c_1+c_2 alpha+c_3 alpha^2 in QQ(alpha).

Find ${c}_{1},{c}_{2},{c}_{3}\in \mathbb{Q}$ such that $\left(1+{\alpha }^{4}{\right)}^{-1}={c}_{1}+{c}_{2}\alpha +{c}_{3}{\alpha }^{2}$
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Brenton Gay
Write $\beta =1+{\alpha }^{4}$ as a linear combination of 1, $\alpha$, ${\alpha }^{2}$ using that ${\alpha }^{3}=-1-\alpha$
Consider the map $x↦\beta x$. Write the matrix of this map with respect to the basis 1, $\alpha$, ${\alpha }^{2}$
Find the inverse matrix and apply to $\left(1,0,0\right)$ to find $\beta$ as a linear combination of 1, $\alpha$, ${\alpha }^{2}$