Given the polynomial $x^4 +ax^2 +b$ I have to determine if it is reducible or not in $Q$. Firstly it hasn't roots, so the only possibility is a factorization between two polynomial of second degree. Being a biquadratic polynomial (I mean a polynomial of fourth degree without the factors of first and third degree) I know that it is equal to $t^2 +at +b$, $t = x^2$ so, found a square ($\alpha$) I know the other ones are $-\alpha , \overline{\alpha}, \overline{-\alpha}$ so I can rewrite the polynomial as $(x-\alpha)(x+\alpha)(x-\overline{\alpha})(x+ \overline{\alpha})$. However this factorization is wrong because the squares aren't in $Q$ so I have to combine the squares so as to obtain a factors in $Q$.
If this method doesn't work (I don't find a combination of square in $Q$) it means that the polynomial is irreducible?