1
$\begingroup$

Take the polynomial $x^4 + 10x^2 + 1$. Is this irreducible over $\mathbb{Q}$? If so, what is the best way to show this? I know it can be rewritten: $$ (x^2 + 5)^2 -24 $$ Which can be simplified over the irrationals, but not over $\mathbb{Q}$. How could I use this to show irreducibility? (if it has anything to do with it)

$\endgroup$
  • $\begingroup$ Here is a key $\endgroup$ – Bumblebee Feb 10 at 21:19
  • 2
    $\begingroup$ You can find the roots of your polynomial by doing $x^2 =t$. Then if non of those 4 roots are rational then the only posibility for the polynomial to be reducible is by a grade 2 polynomial which has to have 2 of the roots you found as solution. Showing that those polynomial are not in $Q$ ends the proof. $\endgroup$ – JoseSquare Feb 10 at 21:22
3
$\begingroup$

Hint: After checking there are no rational roots, you can look for factors of the form $x^2 + a x \pm 1$. Note also that if $p(x)$ is a factor, so is $p(-x)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.