My question is motivated from celebrated Hasse-Minkowski local-global principle:
A quadratic form over rational number $f $ has a solution in $\mathbb{Q}$ if and only if $f$ has solution in $\mathbb{R}$ and $\mathbb{Q}_p$.
I am curious about this principle is effective or not in higher degree polynomial, but with only one indeterminate (or homogeneous polynomial with two indeterminate).
By using elementary knowledge about quadratic residue and Hensel's lemma, I found $f(x)=(x^2-2)(x^2-17)(x^2-34)$ has solution in $\mathbb{Q}_p$ and $\mathbb{R}$ (and especially in $\mathbb{Z}/n\mathbb{Z}$), but it has no solution in rational numbers.
My question is this: Is minimal degree of polynomial of one indeterminate which contradicts to local-global principle is 6?
Also, in similar intuition, is there a polynomial irreducible in $\mathbb{Q}$ which is reducible in every $\mathbb{F}_p$ (or $\mathbb{Q}_p$)?