The Stacks project

Exercise 111.29.6. Unsolved problems. They may be really hard or they may be easy. I don't know.

1. Is there any $f \in {\mathbf Z}[x, u]$ such that $f_ p$ is irreducible for an infinite number of $p$? (Hint: Yes, this happens for $f(x, u) = u - x - 1$ and also for $f(x, u) = u^2 - x^2 + 1$.)

2. Let $f \in {\mathbf Z}[x, u]$ nonzero, and suppose $\deg _ x(f_ p) = dp + d'$ for all large $p$. (In other words $\deg _ u(f) = d$ and the coefficient $c$ of $u^ d$ in $f$ has $\deg _ x(c) = d'$.) Suppose we can write $d = d_1 + d_2$ and $d' = d'_1 + d'_2$ with $d_1, d_2 > 0$ and $d'_1, d'_2 \geq 0$ such that for all sufficiently large $p$ there exists a factorization

   \[ f_ p = f_{1, p} f_{2, p} \]

   with $\deg _ x(f_{1, p}) = d_1p + d'_1$. Is it true that $f$ comes about via a norm construction as in Exercise 4? (More precisely, are there a $h$ and $g$ such that $Norm_{B_ p/A_ p}(g_ p)$ divides $f_ p$ for all $p >> 0$.)

3. Analogous question to the one in (b) but now with $f \in {\mathbf Z}[x_1, x_2, u_1, u_2]$ irreducible and just assuming that $f_ p(x_1, x_2) = f(x_1, x_2, x_1^ p, x_2^ p) \bmod p$ factors for all $p >> 0$.