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What is the simplest radical expression of some root $a \in \overline{\mathbb{F}_5}$ of the polynomial $x^3+x+1 \in \mathbb{F}_5[x]$? I wonder if one can simplify the general formulas in this special case. I think $\sqrt[3]{\sqrt{2} - 3} - \sqrt[3]{\sqrt{2} + 3}$ is ok for the beginning, but this minus in the middle and the nested roots make it rather complicated.

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