which of the following polynomials are irreducible in $ \mathbb{Z}[x] $?

Question

which of the following polynomials are irreducible in $ \mathbb{Z}[x] $?

(a) $ x^{4}+10x+5 $,

(b) $ x^{3}-2x+1 $,

(c) $ x^{4}+x^{2}+1$,

(d) $ x^{3}+x+1 $

My approach: Option (a) is true by Einstein's critera.

option (b) is not true since x=1 is a root.

option (c) is true.

(d) option is also true, for $ x^{3}+x+1 $ has no root in $ \mathbb{Z}[x]$

But I need justification. Is there any help ?

Answer 1

Your justifications for a and b seem good.

As given in the comments, c actually factors.

For d when you reduce mod 2, you get a polynomial that has no roots (just by trying 0 and 1). Since it is a cubic, this tells you that it is irreducible mod 2. Hence it is irreducible in $\mathbb Z [x]$.

Answer 2

For $(d)$, use Mod $2$ irreducibility test.

For $(c)$, it is in fact reducible as $x^4+x^2+1=(x^2+x+1)(x^2-x+1)$