<s>[INST] 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 ?
 [/INST] In fact $$(x^2 - x+1)(x^2 + x+1)$$ $$ = (x^2+1 -x)(x^2+1 +x)$$ $$= (x^2+1)^2 -x^2$$ (remember the  $\alpha^2-\beta^2 =(\alpha - \beta)(\alpha + \beta)$ formula? here $\alpha = x^2+1$ and $\beta = -x $) $$ = x^4 + x^2 +1$$
Well, if you really want to factor $x^4+1$, see here. Hope it helps. 
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