Polynomial with a root modulo every prime but not in $\mathbb{Q}$. [duplicate]
Solution 1:
If you just want an easy example of polynomial that has root modulo every prime but not in $\mathbb Q$ — just take e.g. $$ (x^2-2)(x^2-3)(x^2-6) $$ (it has this property since the product of two non-squares mod p is a square mod p).
One more interesting example is $x^8-16$ (standard proof uses quadratic reciprocity).
As for possibility of complete description of all such polynomials — I'm skeptical.