Let $P=P(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0\in{\mathbb Z}[x]$ be a polynomial of degree $n\ge 1$ with integer coefficients, that is $a_i\in{\mathbb Z}$ for $i=0,\ldots,n$ and $a_n\not=0$.

Which of the following is, by definition, equivalent to $P$ being irreducible over the field ${\mathbb Q}$ of rational numbers?