What is the meaning of the following sentence: $P(x)$ and $Q(x)$ have a common factor modulo $p$.

Question

Let $p$ a prime number. Let $P$ and $Q$ two polynomials over finite fields. What is the meaning of the following sentence:

$P(x)$ and $Q(x)$ have a common factor modulo $p$.

Answer 1

Assuming that $P(x), Q(x)$ have coefficients in the field with $p$ elements, this just means that there is a polynomial $f(x)\in \mathbb F_p[x]$ which divides both of them over that field. Generally one requires that $f(x)$ have degree $≥1$.

Example: with $p=17$ we have $x^4+1\equiv (x+2)(x+8)(x+9)(x+15)\pmod {17}$ and of course $x^2+5x+6\equiv (x+2)(x+3)\pmod {17}$. Thus $x^4+1$ and $x^2+5x+6$ have a common factor of $x+2$ modulo ${17}$. Note that they do not have a common factor over $\mathbb Q$, say.

Answer 2

Think of the polynomials $x^2 + 1$ and $x + 1$ as polynomials with coefficients from the two element field $\mathbb{Z}_2$ of integers modulo $2$. Then since there $$ x^2 + 1 = (x + 1)^2 $$ those polynomials have the common factor $x+1$ modulo $2$.