For some problem from my Galois Theory course, I need to prove that the polynomial $X^4-2X^2+4$ is irreducible in $\mathbb{Q}[X]$.
I know it has no roots in $\mathbb{Q}$ (by rational root theorem), but I can't conclude it's irreducible with just that fact because it's degree is $4$, and tried several changes to try to use Eisenstein Criterion but I had no luck.
How can I prove it? Is it really irreducible in $\mathbb{Q}[X]$? Thanks in advance, any help will be appreciated.
Best Answer
If it was reducible, you would be able to write it as $(X^2+aX+b)(X^2-aX+c)$. But$$(X^2+aX+b)(X^2-aX+c)=X^4+(-a^2+b+c)X^2+a(c-b)X+bc.$$So, you would have$$\left\{\begin{array}{l}-a^2+b+c=-2\\a(c-b)=0\\bc=4.\end{array}\right.$$So, $a=0$ or $b=c$. If $a=0$, then $b+c=-2$ and $bc=4$, and you can easuly check that this system has no rational solutions. And if $b=c$, you get $-a^2+2b=-2$ and $b^2=4$. Again, you can check that this system has no rational solutions.