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)

  • $\begingroup$ 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

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 $p(-x)$.


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