Mod p irreducibility test : Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by reducing all the coefficients of $f(x)$ modulo $p$. If $f_1(x)$ is irreducible over $\mathbb Z_p$ and $\deg f_1(x)=\deg f(x)$, then $f(x)$ is irreducible over $\mathbb Q$.
The question states here:
Which of the following polynomials are irreducible in $Z[X]$?
(a)$x^4+10x+5$
(b) $x^3-2x+1$
(c) $x^4+x^2+1$
(d) $x^3+x+1$
Here (a) is irreducible over Q by Eisenstein's criterion, for $p=5$
(b) has root $x=1$ so is reducible
For (d) if I use mod p irreducibility test for $p=2$ then since it is irreducible over $\mathbb{Z_2}$, it is also irreducible over Q.
And Similarly I can say for (c), $x^4+x^2+1$ for prime 2, it is also irreducible over$\mathbb{Z_2}$, so it must also be irreducible over Q.
But since we can write $x^4+x^2+1=(x^2+x+1)(x^2-x+1)$ , this implies it is reducible over $\mathbb{Q}$ and $\mathbb{Z}$.But I got it irreducible by p mod irreducibility test so where I am wrong with p mod test?
