# Irreducibility of the following Polynomial over $\mathbb{Q}$

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)

• Here is a key – Bumblebee Feb 10 at 21:19
• 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. – 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 $$p(-x)$$.