MathNet 2018-part-04 — 181 problems (EN–VI) Question 154

For $k \in \mathbb{Z}$ define the polynomial $F_{k}=X^{4}+2(1-k) X^{2}+(1+k)^{2}$. Determine all values $k \in \mathbb{Z}$ such that $F_{k}$ is irreducible over $\mathbb{Z}$ and reducible over $\mathbb{Z}_{p}$ for every prime $p$.