How could I know that $X^4+1$ is $(X^2+\sqrt 2X+1)(X^2-\sqrt 2X+1)$?

I thought that $X^4+1$ was irreducible, but in fact, $$X^4+1=(X^2+\sqrt 2X+1)(X^2-\sqrt 2X+1).$$

In general, how can I have the intuition of such a factorisation if I don't know it ?

Solution 1:

There's a trick here, that is useful in other circumstances. I will do it over the real numbers $$ x^{4} + 1 = x^{4} + 2 x^{2} + 1 - 2 x^{2} = (x^{2} + 1)^{2} - (\sqrt{2} x)^{2} = (x^{2} + 1 - \sqrt{2} x) (x^{2} + 1 + \sqrt{2} x). $$ So it's just completing the square.

Solution 2:

Hint

You can easily solve $X^4+1=0$ in $\mathbb C$ and identify which product of two monic are in $\mathbb R[X]$.