The Stacks project

Exercise 111.29.5. Let $h, A, B, A_ p, B_ p$ be as in the remark. For $f \in {\mathbf Z}[x, u]$ we define $f_ p(x) = f(x, x^ p) \bmod p \in {\mathbf F}_ p[x]$. For $g \in {\mathbf Z}[y, v]$ we define $g_ p(y) = g(y, y^ p) \bmod p \in {\mathbf F}_ p[y]$.

1. Give an example of a $h$ and $g$ such that there does not exist a $f$ with the property

   $$f_ p = Norm_{B_ p/A_ p}(g_ p).$$

2. Show that for any choice of $h$ and $g$ as above there exists a nonzero $f$ such that for all $p$ we have

   $$Norm_{B_ p/A_ p}(g_ p)\quad \text{divides}\quad f_ p .$$

   If you want you can restrict to the case $h = y^ n$, even with $n = 2$, but it is true in general.

3. Discuss the relevance of this to Exercises 6 and 7 of the previous set.