0
$\begingroup$

Can someone give me a hint about how to prove that a polynomial is irreducible over $\mathbb{Q}(i)$ when we know that this is irreducible over Q?

I have tried to do this by hand (supposing there are factors for the polynomial) but I did not get anything.

Thank you in advance.

  • 1
    That's not true. $x^2 + 1$ is irreducible over $\Bbb Q$, but not over $\Bbb Q(i)$. Unless you have a _specific_ polynomial in mind? – 2017-01-29
  • 0
    What about $x^3-4x+2$? – 2017-01-29
  • 0
    That one has three real roots (you can check that by looking at it's first and second derivative). So the real question is whether these roots lie in $\mathbb{Q}$ or not. So in this case it suffices to consider irreducibility over $\mathbb{Q}$ to get it over $\mathbb{Q}(i)$. – 2017-01-29

0 Answers 0