$$a^{12}+a^6+1 = 0$$
Answer
$$ \begin{matrix}a_1 = 0.76604+0.64279i & a_2 = 0.76604-0.64279i & a_3 = -0.76604+0.64279i \\[1 em] a_4 = -0.76604-0.64279i & a_5 = -0.17365+0.98481i & a_6 = -0.17365-0.98481i \\[1 em] a_7 = 0.93969+0.34202i & a_8 = 0.93969-0.34202i & a_9 = 0.17365+0.98481i \\[1 em] a_10 = 0.17365-0.98481i & a_11 = -0.93969+0.34202i & a_12 = -0.93969-0.34202i \end{matrix} $$
Explanation
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using Newton method.
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