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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.

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View Exercise 111.29.2 as pdf