Show that if a polynomial f(x) in Z[x] of degree n has no rational root, but for some prime p, its reduction mod p has irreducible factors of degree 1 and n-1, then f is irreducible
Please, help

Question

Show that if a polynomial f(x) in Z[x] of degree n has no rational root, but for some prime p, its reduction mod p has irreducible factors of degree 1 and n-1, then f is irreducible
Please, help

loser66 · Answer

My attempt: 
$f(x) = a_0 + a_1x+a_2x^2+.....+a_nx^n$

loser66 · Answer

let g(x) is its reduction mod p, hence 
$g(x)=[a_0]+[a_1]x+.....+[a_n]x^n$

loser66 · Answer

I can set that because g(x) has 2 factors with degree 1  and n-1, that is degree of g(x) is n

loser66 · Answer

And that fact gives us $a_n$ stays as it is in g(x)

loser66 · Answer

but after "reduction modulo p" $a_n$ doesn't change, hence p is not divide $a_n$
My strategy is to try to apply Eisenstein's Criterion.

loser66 · Answer

But I don't know how to do next.

miracrown · Answer

Me neither, sorry. I haven't learnt this material as yet, so I am completely blank.

loser66 · Answer

Thanks for reply anyway. :)

miracrown · Answer

:)