there are only five solvable quintic equations of the form $x^5+ax^2+b=0,~~a,b \in \mathbb{Q}$:$$x^5-2s^3x^2-\frac{s^5}{5}=0 $$
$$ x^5-100s^3x^2-1000s^5=0 $$
$$x^5-5s^3x^2-3s^5=0 $$
$$x^5-5s^3x^2+15s^5=0 $$
$$ x^5-25s^3x^2-300s^5=0 $$
设 `a`, `b`, `c` 是 `x^3 + 3 x - 1 = 0` 的三根,即 `a + b + c = 0`, `a b + b c + c a = 3`, `a b c = 1`,则
\[
\bigl(x^2 - (a^2 + 2) x + a + 1\bigr) \bigl(x^2 - (b^2 + 2) x + b + 1\bigr) \bigl(x^2 - (c^2 + 2) x + c + 1\bigr)
= x^6 - 3 x + 5,
\]`a`, `b`, `c` 的具体值为 `-\omega\sqrt[3]{\frac{\sqrt{5}-1}{2}}+\omega^2\sqrt[3]{\frac{\sqrt{5}+1}{2}}`,其中 `\omega` 为任意三次单位根。