<s>[INST] How can I systematically find the roots of $ x^4 + 1?$ Is there some algorithm? 
Possible Duplicate:
How to find the root of $x^4 +1$ 

What algorithms can be used for finding all roots of the given polynomial:
\begin{equation}
    x^4 + 1 = 0
\end{equation}
 [/INST] $$x^4 + 1 = (x^2 + 1)^2 - 2x^2 = \\
(x^2+1+\sqrt2  x) (x^2+1-\sqrt2 x) =\\
(x^2 - x\sqrt 2+1)(x^2  + x\sqrt2+1)=0$$


*

*$x^2 - x\sqrt 2+1=0$
$$x_{1}={\sqrt2+\sqrt{-2}\over2}={\sqrt2\over2}(1+i)$$
$$x_{2}={\sqrt2-\sqrt{-2}\over2}={\sqrt2\over2}(1-i)$$

*$x^2  + x\sqrt2+1=0$


$$x_{3}={-\sqrt2+\sqrt{-2}\over2}=-{\sqrt2\over2}(1-i)$$
$$x_{4}={-\sqrt2-\sqrt{-2}\over2}=-{\sqrt2\over2}(1+i)$$
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