Find an irreducible polynomial in this case, if it exists

Question

Is there a second degree polynomial $P(x)\in\mathbb{Q[x]}$ so that it is irreducible in $\mathbb{Q}$ but reducible in $\mathbb{Q[i]}\in\Bbb{C}$?

I know that an irreducible polynomial cannot be factored into the product of two non-constant polynomials and it depends on where the coefficients are taken from. What can I say in this case? Any help or hint is much appreciated.

Answer 1

A second degree polynomial is reducible in any field means the roots of the polynomial exist in that Field. So you need a second degree polynomial which is irreducible in $\mathbb{Q[x]}$ but in $\mathbb{Q[i]}$. So overall you just need to find a equations whose roots are not rational but either real or complex . So you can find many examples over it. Very simple example i can give you is

$X^2+ 1\in\mathbb{Q[x]}$ but irreducible there.

But

it's reducible in $\mathbb{C[x]}$ as into $(x-i)(x+i)$ .

That is very simple example as you ask for it but you can stress on your mind and can think many of them.