common roots of polynomials in Galois fields and the complex numbers

Question

Is the following true or false? If two polynomials of degree $n>2$ have a common root in $\mathbb{Z}/p\mathbb{Z}$, where $p$ is prime and $n|(p-1)$, then they have a common root in $\mathbb{C}$? If so, why?

(I am not a student and this is not an exam question.)

Answer 1

The result is false:-

The polynomials $x^3+1$ and $x^3+5x$ have common root $3$ over $\mathbb{Z}/7\mathbb{Z}$ but no common root over $\mathbb{C}$.