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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.

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    $\begingroup$ A polynomial of degree $2$ is reducible over a field if and only if it has a root in that field. (Do you understand why?). If you know that, it should be really easy to find an example. $\endgroup$
    – Mark
    Jun 8 at 15:51
  • $\begingroup$ Thank you for the comment. No, unfortunately I don't know why since this is the first time we are doing polynomial over rings or fields. And also, since Q isn't a field, could you please explain me also the connection here? $\endgroup$
    – katrin
    Jun 8 at 15:55
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    $\begingroup$ Why $\mathbb{Q}$ isn't a field? It is. Anyway, if a polynomial $f\in F[x]$ is reducible then it is a product $f=gh$ of two polynomials of lower degree. But if the degree of $f$ is $2$ then the polynomials $g,h$ must have degree $1$. So $g$ has the form $g(x)=ax+b$ for some $a,b\in F$ with $a\ne 0$. So $\frac{-b}{a}\in F$ is a root of $g$, and hence a root of $f$. $\endgroup$
    – Mark
    Jun 8 at 16:03
  • $\begingroup$ So if a monic 2nd degree polynomial is reducible, it has a monic factor of degree 1. That gives a root. $\endgroup$
    – hardmath
    Jun 8 at 16:04
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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.

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  • $\begingroup$ is there a similar rule for third degree polynomials? $\endgroup$
    – katrin
    Jun 8 at 19:48
  • $\begingroup$ @kay yes it's ALWAYS true for degree 2 or 3 . But gernally not for higher degree's than 3. $\endgroup$ Jun 8 at 20:38

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