Question

x^8 – x^4 + 1001 ≡ 0 (mod 539)

Answer 1

\(539\;=\;11\cdot 7^{2}\)

\(1001\;=\;11\cdot 7\cdot 13\)

Answer 2

I've got to go soon, but you can simplify to \[x^8-x^4+462\equiv 0\pmod{539}\]Then, you can use the quadratic formula to solve for \(x^4\). We get \[\frac{1\pm\sqrt{-1847}}{2}\]However, we're still working in modulo 539, so \(\sqrt{-1847}\equiv \sqrt{309}\pmod{539}\). So you need to solve the congruence \(y^2=309 \pmod{539}\).

Answer 3

That will get you the square root in the quadratic formula, and you can solve from there.