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I found the following question in S.-T. Yau College Student Mathematics Contest 2013:

  • Find a polynomial $f(x)$ with integer coefficients which has a root over $\mathbb{F}_p$ for each prime $p$ but has not root over $\mathbb{Q}$.
  • What is the smallest possible degree of $f$ $?$

Prior to that I searched for a polynomial with integer coeffients which is irreducible in $\mathbb{Q}[x]$ but is reducible modulo every prime $p$. All roots of $x^4+1$ are roots of $x^{p^2}-x$.Which means any root of $x^4+1$ is at most of degree 2 over $\mathbb{F}_p$,and $x^4+1$ cannot be irreducible over $\mathbb{F}_p$.

But irreducible not not equivalent to have no root.How can I use the ideas in this question$?$

Dual Rray
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    Asked and answered here earlier. See, e.g., https://math.stackexchange.com/questions/608919/is-it-true-that-if-fx-has-a-linear-factor-over-mathbbf-p-for-every-prim and https://math.stackexchange.com/questions/1086123/polynomial-with-a-root-modulo-every-prime-but-not-in-mathbbq and others linked to those. – Gerry Myerson Apr 25 '25 at 04:08
  • While the knowledge from all those links put together can answer this question, this is not a straightforward duplicate, at least not for beginners. For example, one needs to reduce the problem from $\mathbb Q[x]$ to $\mathbb Z[x]$. As far as I can tell, many similar posts were similarly voted as duplicates, while the answer was never explicitly given. It's a shame that this question is closed and every time such a problem is raised, it will only make it easier to be closed again. – Just a user Apr 25 '25 at 04:53
  • @Just, you can re-ask the question, pointing out just how it differs from each of the dozen other similar questions, and post the answer you get by combining elements from the earlier answers. That would be a service to the community. – Gerry Myerson Apr 26 '25 at 04:07

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