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

Question

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? – Angina Seng Apr 20 '19 at 11:56 — Angina Seng, Apr 20 '19 at 11:56
Just look for a simple quadratic. Try the form $x^2+a$ . – lulu Apr 20 '19 at 11:59 — lulu, Apr 20 '19 at 11:59
Have you tried some simple quadratics? – rogerl Apr 20 '19 at 11:59 — rogerl, Apr 20 '19 at 11:59

score 1 · Answer 1 · answered Apr 20 '19 at 12:02

1

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

answered Apr 20 '19 at 12:02

little o

4,853

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

1 Answers1