Decomposition of polynomials like $1+x^4$ What is the "trick" to get from $1+x^4$ to $$(x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1)?$$ Of course I can calculate it's true, but I don't understand what steps to take to get from one to the other.
Next to this specific question, I am also looking for the general rules for such decompositions?
Cheers!

Use the binomial formula $x^4+1=(x^2+i)(x^2-i)$ and compute the roots of the factors
$$x_{1,2} = \pm \tfrac{1}{2}\sqrt{2}(1+i)$$
$$x_{3,4} = \pm \tfrac{1}{2}\sqrt{2}(1-i)$$
Now multiply 
$$(x-x_1)(x-x_3) = x^2 - \sqrt{2}x +1$$
$$(x-x_2)(x-x_4) = x^2 + \sqrt{2}x +1$$
