Question 57
Let $\mathbb{Z}[x]$ be the set of polynomials with integer coefficients. Let $q$ be a prime number. A polynomial $P \in \mathbb{Z}[x]$ is called **q-reducible** if there exists $f_q, g_q \in \mathbb{Z}[x]$ such that $f_q, g_q$ have at least one non-constant coefficient which is not divisible by $q$, and all the coefficients of polynomial $Q(x) = P(x) - f_q(x)g_q(x)$ are divisible by $q$.
a. Prove that there exists infinitely many prime numbers $q$ such that the following polynomial is $q$-reducible
$$
P(x) = x^4 - 2x^3 + 3x^2 - 2x - 5
$$
b. Does there exist a polynomial $P \in \mathbb{Z}[x]$ which is $q$-reducible for every prime number $q$ but is irreducible over $\mathbb{Z}[x]$?
a. Prove that there exists infinitely many prime numbers $q$ such that the following polynomial is $q$-reducible
$$
P(x) = x^4 - 2x^3 + 3x^2 - 2x - 5
$$
b. Does there exist a polynomial $P \in \mathbb{Z}[x]$ which is $q$-reducible for every prime number $q$ but is irreducible over $\mathbb{Z}[x]$?