The Stacks project

111.29 Infinitely many primes

A section with a collection of strange questions on rings where infinitely many primes are not invertible.

Exercise 111.29.1. Give an example of a finite type ${\mathbf Z}$-algebra $R$ with the following two properties:

  1. There is no ring map $R \to {\mathbf Q}$.

  2. For every prime $p$ there exists a maximal ideal ${\mathfrak m} \subset R$ such that $R/{\mathfrak m} \cong {\mathbf F}_ p$.

Exercise 111.29.2. For $f \in {\mathbf Z}[x, u]$ we define $f_ p(x) = f(x, x^ p) \bmod p \in {\mathbf F}_ p[x]$. Give an example of an $f \in {\mathbf Z}[x, u]$ such that the following two properties hold:

  1. There exist infinitely many $p$ such that $f_ p$ does not have a zero in ${\mathbf F}_ p$.

  2. For all $p >> 0$ the polynomial $f_ p$ either has a linear or a quadratic factor.

Exercise 111.29.3. For $f \in {\mathbf Z}[x, y, u, v]$ we define $f_ p(x, y) = f(x, y, x^ p, y^ p) \bmod p \in {\mathbf F}_ p[x, y]$. Give an “interesting” example of an $f$ such that $f_ p$ is reducible for all $p >> 0$. For example, $f = xv-yu$ with $f_ p = xy^ p-x^ py = xy(x^{p-1}-y^{p-1})$ is “uninteresting”; any $f$ depending only on $x, u$ is “uninteresting”, etc.

Remark 111.29.4. Let $h \in {\mathbf Z}[y]$ be a monic polynomial of degree $d$. Then:

  1. The map $A = {\mathbf Z}[x] \to B ={\mathbf Z}[y]$, $x \mapsto h$ is finite locally free of rank $d$.

  2. For all primes $p$ the map $A_ p = {\mathbf F}_ p[x]\to B_ p = {\mathbf F}_ p[y]$, $y \mapsto h(y) \bmod p$ is finite locally free of rank $d$.

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.

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 $p >> 0$.

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