# Finding an irreducible polynomial in $\mathbb{Z}[x]$ such that it is reducible modulo 2,3 and 5. [duplicate]

The problem is finding an irreducible polynomial in $$\mathbb{Z}[x]$$ such that it is reducible modulo 2,3 and 5. I can't find anything, any help is appreciated. (Is there some general strategy for doing this?)

• Chinese remainder theorem? Apr 20, 2019 at 11:56
• Just look for a simple quadratic. Try the form $x^2+a$ .
– lulu
Apr 20, 2019 at 11:59
• Have you tried some simple quadratics? Apr 20, 2019 at 11:59

Hint $$:$$ Take $$f(x)=x^4+1.$$ Then it is reducible modulo every prime $$p.$$ But it is irreducible in $$\Bbb Z[x].$$