0

Let $f(x) = x^4 -10x^2 + 1$ a polynomial. Prove that $f$ has a factorization over $(\Bbb Z/p\Bbb Z)[X]$, $\forall$p prime number.

I have proved that this polynomial is irreducible over $\Bbb Z[X]$ using rational root theorem, but factor over $(\Bbb Z/2\Bbb Z)[X]$ because of $\overline 1$ is a root.

Now I'm not able to prove the generalization.

Oriol Sors
  • 83
  • How does the rational root theorem apply here? I can see how that shows there is no rational root, but it still might factor as the product of two irreducible quadratics. – lulu Sep 24 '19 at 15:09
  • $f(x)= ((x+\sqrt{a})^2-b) ((x-\sqrt{a})^2-b)$ so $f$ is not irreducible over $\Bbb{F}_p(\sqrt{a})$ – reuns Sep 24 '19 at 15:10
  • @lulu Yes, when I proved that there is no rational root, then I assume that $f$ factor as the product of two irreducible quadratics, which ends with a contradiction because the coefficients got are not rational. – Oriol Sors Sep 24 '19 at 15:36

0 Answers0