I was looking if $X^4 + 2$ was irreducible in $\mathbb{R} \left[ X\right] $, $\mathbb{Q} \left[ X\right] $ and $\mathbb{Z_{7}} \left[ X\right] $. It's easy to prove that in $\mathbb{R} \left[ X\right] $ is irreducible due to the fact that has $2$ complex roots.
The previous reason is applicable for $\mathbb{Q} \left[ X\right] $, but in $\mathbb{Z_{7}} \left[ X\right] $ I don't know how to do it. Any ideas?
On
Note that all roots are complex does not imply irreducibility. In this case, that polynomial can be written as a product of two quadratics over $\Bbb R$
The only polynomials irreducible over $\Bbb R$ is of degree one or two, since $[\Bbb C : \Bbb R]=2$, so your polynomial is reducible over $\Bbb R$
Eisenstein criterion shows it is irreducible over $\Bbb Q$
For the case $\mathbb{Z}_7$, notice that $$X^4+2=(X^2+X+4)(X^2-X+4)$$